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DECLARATION OF RONNIE ELLIOTT

Ronnie Elliott, being fixst duly sworn upon her oath, deposes and says:

1. I am over the age of eighteen and am competent to testify as to all the

matters set forth herein and would so testify if called upon to do so.

2. I have personal knowledge of the matters set forth herein. My personal

knowledge is based upon my observations and personal participation in the events

described below.

3. I was employed with Maricopa County Community College District

("MCCCD") from the early 1990s through January 2010. I served as Phoenix College's

("PC") Vice President of Administrative Services from approximately 20Q7 through

January 2010. On January 30, 2010 I voluntarily retixed after an approximately 20 year

long career with MCCCD. Following my retirement, Paul DeRose took over my

responsibilities as PC's Vice President of Administrative Services. I now reside in

Buffalo, Missouri.

4. In my former role as PC Vice President of Administrative Services, I was

responsible for supervising our various vendor services including custodians, information

technology ("IT"} personnel, ar~d copy service personnel. I also met and interacted with

numerous faculty members. One such faculty member was Math Professor Dr.

Cleopatria Martinez.

5. Although I was not involved with the PC's IT department's day-to-day

tasks, I oversaw the IT Departnnent's general operations and computer maintenance.

Specifically, the IT Department is responsible for maintaining PC's computer systems

and ensuring: (1) that no unauthorized software has been downloaded that could damage

PC computers; and (2) that PC systems are not being accessed by unauthorized

individuals. Tn my former role as Vice President of Administrative Services, the IT

Department reported to me and any suspicious findings were forwarded to my attention.

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6. In or around Fall of 2009, while the IT department was performing

routine conrzputer maintaince, it was discovered that an illegal file sharing program called

LimeWire had been installed on Dr. Martinet's laptop computer. It was fiu-ther

discovered that a high school level homework assignment authored by an individual with

the last name "Martinez" was present on Dr. Martinet's MCCCD issued Laptop. In

accordance with the IT Department's normal practice, the findings were immediately

forwarded to my attention. Upon my review of the findings, I concluded that Dr.

Martinez had improperly permitted an unauthorized user to access her MCCCD laptop

computer. Since I was not responsible for disciplining faculty members, I notified Vice

President of Academic Affairs Cassandra Kakar and PC 'resident Anna Solley of the

findings.

7. A few weeks later, on or around September 24, 2009, I learned that PC

custodian Verna Daubney, had found dozens of manila folders containing confidential

faculty applicant information (including social security numbers and school transcripts)

in Dr. Ivlartinez's office garbage can. Dr. Martinet's disposal of these materials in this

way violated MCCCD's policies and procedures which require that applicant materials be

disposed of in a secure manner. I reported the findings to PC's Vice President of

Academic Affairs, Calandra Kakar, and PC President Atuia Solley.

$. In January of 2010, our copy service provider, IKON, notified me that that

Dr. Martinez had submitted a set of suspicious copy requests. Apparently, Dr. Martinez

asked IKON to copy approximately 30 sets of a 73 page document titled "2009 MAT

182." Dr. Martinez had also submitted a similar request for her M~1T I87 course. IKON

was concerned that the documents appeared to contain math problems taken directly from

textbooks and wanted my guidance regarding whether to process the print jobs. At zny

request, IKON forwarded a copy of the "2009 MAT 182" document to my attention. A

true and correct copy of these materials are attached as Exhibit A.

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9. After reviewing the materials I received from IKON, I concluded that the

materials presented a serious risk of copyright infringement and forwarded the materials

to PC Vice President of Academic Affairs, Cassandra Kakar, and PC President Anna

Solley. Drs. Kakar and Solley reached the same conclusion and instructed me to bring

the matter to the attention of Dr. Martinez. Accordingly, on or around January 12, 2010,

I sent an eznai] to Dr. Martinez explaining that PC had "red flagged" items Dr. Martinez

had submitted for printing for the Fall 2009 and Spring 2010 serraesters and that we were

concerzied that the materials tray subject PC to a claim of copyright infringement. A true

and correct copy of nny January 12, 2010 email to Dr. Martinez is attached as Exhibit B.

10. Much to my surprise, rather than apologize for her behavior and remove

any copyright portions of her materials, Dr. Martinez began asking me (and others within

MCCCD's administration) if she could "fix" the copyright issues by changing one or two

numbers from problems that were copied directly from copyright protected textbooks.

Dr, Martinet's attempts to elicit advice fronrz ine (and others) on how to evade applicable

copyright laws made me feel extremely uncomfortable. It also became quite apparent to

me that Dr. Martinez had no intention of abiding by copyright laws and regulations.

11. On January 26, 2410, X sent Dr. Martinez another email outlining

MCCCD's concerns regarding her suspected copyright violations. Specifically, X

explained that I had solicited the advice of district counsel, Margaret McConnell

regarding copyright infringement and fair use. I further explained that we were

concerned that Dr. Martinet's actions could subject MCCCD to a copyright infringement

claim because we were able to identify at Ieast 10 instances where problems had been

copied directly from the Sullivan and Sullivan Precalculs textbook. A true and correct

copy of rrzy January 26, 2410 email to Dr. Martinez is attached as E~ibit C. I retired

from MCCCD several days later on January 30, 2010.

12. My decisions to report Dr. Martinet's above referenced actions to Drs.

Solley and Kakar were not based on any discriminatory or retaliatory feelings towards

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Dr. Ma~-iinez. To the contrary, I felt obligated to repoi-c my concerns regarding Dr.

Martinez to PC administration in light of the potential legal liability that PC could have

faced as a result of Dr. Martinez's actions.

Pursuant to the laws of the State of Arizona, 1 declare under penalty of perjury that

the Foregoing is I~ue and correct.

Executed this 9th day of October, 2013 iii. $uffalo, Missouri.

V'

RONNIE ELLIOTT

MCCCD/Martinez 01245

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Exhibit A

MCCCD/M~rtinez 01246

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~ P~oenz~ College

Fall 2009

~ Tr~go~o~e~r~ Sec~Tor~

Lec~~ ~[o~s

6 EXHtB17 N0. ,s.LL.~--•

3

0

C

Dr. C1eapar~a Mc~ne2

~~

MCCCDIMartinez00451

MCCCD/Martinez 01247

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MA718Z Tri~n~mctry Pagc 1 of 73 Spann 2010

f~` ~ -- ~'~~Q~cQt~e~t

1.1 fln~es an d Ehe=r hSeasure . Basic Angles, t~egrees° Minute' Seconds", Degrrees,

Radians, arc length, area of a sector, linear speed, angular speed

'~acabiilary: Angle, ray, Iizte, segment standaxdposition of an angle mitiat szde,

A se ent is a sefi Qf points with a startiztg point and an ending p~in~.

A Tine is a set of points that go famver in two dizectinns. ~

A. ~ is a set of points that stazE aE one gozctt then go foz~ver.

An an e is two.ra~rs with the same star•Euig painE.

the vertex is the point at foie corm of the angle, r~~tsta~ A

The initia.~ side is the ray where tTze angle starts..

• The terminal side is the ray where tt~e ankle ends. C

A ~asifizve angle opens counter-clockv~rise. °mac« ~;ti~is~d~ g .

A negative an_gIe opens clockwise_

The name of t3xe an e canbe tote vertex, like angle C ie. LC orby using three letters

wi$t the vertex in the middle i:e. LACB of LBCA.

Standard~osition h~.s the vert~c at the origin and the initial side on the positive x-axis,

1 he size a£ asp an~ e is measutec~ by degz~es, A fuS1 rntatian is 36D° i.e. 36Q degrees.

I)e~rees ° Minutes' Seconds" are used to desrn'be the size of an angle,

One de~iee, ~rrritten 1 °, reFireaerrts - 6 3 0#' a fuI1 natation,

One minute, wriften 1', represents 60 of a degree.

A degree can be divided mto 6Q pants cared 6Q zniziutes i.e. 60`.

One second wrztfien 1`, represents 60 of a minute.

A degree can be divided into 6Q pans called 6Q seconds ie_ 60".

An acute angle has between a degrees and 90 degrees.

A ri tan e has exactly 9Q degrees.

An aF~Ense an~Ie has between 40 degrc:~s and 18.4 degrees.

t~ strai.ght angle has ~actl~ 1$Q degrees.

A reflexive an~;ie has between 1£30 degrees and 3b0 degrees,

A complete rotation gives an a-zgle measu.ting 36Q degz~ees.

Camptementary an es are two angles which add up to 9Q degzees.

~~ementary an~Ies are two angles which add up to 1$Q degrees,

MCCCD/MartinezOQ452

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MA.TI82 Tri~oaometry P~ e 2 of 73

Nate the right triangle inscn7~ed in the cizrle.

The teLnmlal side of ~e angle is not only the hypot~

o€ the right biangIe but it is also fine radius

of the circle.

terminal side

vPitex initial side

—~- --!

1 degree° = 60 minuEes = 5g minutes + 6t1 seconds

r1

2 minute' — 7.' = f —~o

~60~

k ~ ~ ~ o

1 second" --= Z " = (bQ~ 60 ~ 60

1 mi_tcute` = 6Q seconds

Spring 2010

Qz~~zge B~`ees. lY~~ttutes--~eeonds ineo I?eAre~s zn decc_imnal form.. _

Give- the de~J.mal a~swPs accurate to two decimal places.

example: Cliaztge ~0°6' 2J " inco degrees. Exasngle: Change G1 °42` 2I" iz~m degrees.

Answer: 

Answer_ .

50 ° ~- 6' ~- 21"

0 r

`0

5Q° ~- ~ ~6~ ( j' li ~6D n 6Q~ 

Q

J

0

~o { (lOJ } ~l7 (3.2060) o

~ o

50 ° ~t- .1 ° + —

1200

5D ° + 1 ° t- DOS833

50.105833 ~ or 50.11

Mc~CD/Martinez00453

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M?.T182 Tri~onomehy Pa;e 3 of 73 Sgring2010

~xa~t~e ~ esS into Ilegrees° Mi~cufes ` Seccrrcds"

F_xau,ple: Ch3ilge 21256 ° into Degrees °Minutes'Secorids" ~xamgle: Ckktnge 6124 into TAMS

lu~scver_ ,Aiuwer:

27. ~ f 256'

z~~ + a~~ (6ta~,

zl ° + xs.~

27 ° + 1s~ -~ .3fi ~ 

.

21° + 15 ` +.36(60)„

21 ° I5' 22" .

In many applications, IiTce describing the enact location of a sdar or the enact posidact o£

a boat at sea, angles mea4ured in degrnes~ uunutes, and sernnds are used. These are

usually changed to degrees in decimal fornti

1.1 Antes and their Measure ~~azx~e~ai'k

~..ial page 7 problems 1-21 odd, 23-28

Sullivan:

1n problems 11-2I, draw each angle.

11. 30' 13. 135' I5. 45D' 17. '~ 3 19. —'~ 21 16n

4 6 3

Convert each angle to a decimal in degrees. Round youi answer to tv✓o decimal places.

23. 40'10'25^ 25. 1'2'3" 27. 9'9'9"

Convert each angle to D'M'S" farm. Rotmd yoiu' answer to the neatest second.

29. 40.32' 31. 18,255' X33. 19.99"

MCCCD/Martinez~0454

MCCCD/Martinez 01250

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l~latb 1$2--Spring 2010

~~~~ ~'.~~~axa~r~~z~ ~-a~n~.

Due Date Chapter 3 — T'ri~onnme~ric Identities Page

3.I.14 TrigYmome~-lc Identities (30) — 53

3.215 Sum & Drfferences Formulas (35) 57

3.3.16 Double-angle &Power Reduang Form~7ias (35) -- 63

3.4. Test on Trigonometric Identities --(no homework) 67

~1~7{~F' ~ •-- f~.F~F1RI~G~E~ ~'i''i~FYfOY~~{7"4~

4.1.17 Trigonometric Equations (56) ------- 69

4.2.18 Applications isivoEving Right Triangles (18)-- — 73

4,3.19 The Law of Sines (1~ --- 75

4.4.20 The Law of Cosines (16) 79

1l~AT 1~2 Final Exam '6~ednesday,May 12

10:40 -11:50 am xn ~ Room 5210

MGCCDlMartinez00455

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;:

~,.

y:

~==:

,:r•

;,,,.

v'

~r;

i.'i•

r.

.i

Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 32 of 122

MAT182 Trigonometry k~e 4 of. 73

1-2 Ames i~ De~ees ar~d Az~g~es ~zz R:adians

Tn calculus, angles are measured using radiwcs.

A cenfral aYtgie is an angle whose vertex is at ire center of a circle.

f?ne radian is the measure of a central angle B that cuts an arc s

which is as long as tie radius r of the circle: s = r

That angle is ane radian, around 57 ° .

Springy 2Q 10

Ijthe mzgle is X radiant, lher: the length of the arc is also X radir~rc:

ie when the arc tengtlt =radius, then the angle is I radian or abotrl 57°~

If an angle Bis in degrees, use the number and the degree symbol when wriling atrigonometric

function like sin 30' and tan 45' . Tf an angle B is r~easured in ~aclians, then no symbol is used to

represcni radians when writing a triganometrac funcfion like cosec and sec 3 .

Greek letters: a alpha ~3 beta B theta ~ phi m omega n= pi = 3.14159265 radians

1 revolution = 2 ~r radians

1 revolution= ~t radiane

2

Divide ~c by 4 and label the angle.

lc ~r 3_r 4x

t 4 4 4

Divide n by 3 then divide ~r by 6.

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MAT182 Trigonometry Page 5 of r

Use the circle to mark the degree and radian meastue of common an;les.

Divide the cixcle into mu3tiples of 30 ~ , t3~en divide the circle into multiples of 45 ° .

Transfer this information to the table provided.

s~o zo i o

Degrees 0° 30° 45° 60° 9Q° 120 135° 150° 180°

Radians

Degrees 210° 225° 240 ~.70° 300 315 330 .360°

C~'taxtee I?ePt'ees i~xtaradi~.ns;

1&0 degrees = n radians

Y degz'ee = Z80 radiazls

Solve foc 1 degzee by

diy~ding both sides by 180

Chazt~e R~Iifrrzs inEa degrees Solve for 1 radian by

~' rdG~ians = 18~ degrees dividing bokh sides by n

1 radian = I8~ degrees ~ 573°

n

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MAT182 Trigonometry Pagc 6 of 73

~cample: Convert 60 ~ inYo radians.

Answer: 60° = 60(1 degree)= 1~ 0~ = 3 radians

Example: Change 107 into radians.

E~mple: Convert 6 radians into degrees.

Answer ~ radians = ~ ~1 radian =~ D 180 degrees=3o degrees ie 30°

6 6 6 ~r

F~cample: CUange 3 radians into degrees.

x

1 2 Andes in ~e~and Antes ~t RaE[ians H~meHraxk

Convert each angle in de~ees to radians. Express your answer as a mul~ple of ~r .

35. 30' 37. Z40' 39. --b0' 41. 180' 43. —135` 45_ —90"

Convent each angle in radians to degrees,

4'7, ~ 49, —' S~- 51. ~ 53, n 55. — ~ 57_ —'~

3 4 2. 12 2 6

Convert each angle in degrees to radians. Round an_svrer to two decimal places.

54. 17' 61. —40' .. 63. 125'

Convert each angle Sn radians to degzees. Round answer to two decimal places.

b5. 3.14 67.2 G9. 632

Spring 2D 10

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MA7182 Trigonometry Page 7 of 73

I~,Axcl~x~~h~Area, anc~'(~eloczt~

Spring 2014

Arc LenQfh; "The len~h of an arc ~ made by a central angle i.s found by multiplyzug the

i~adnis times the an e measure. ~ 7{~

L of Big circle ~ Bid arc length f

L of sma1C circle "1 " ~ small arc length "r".

If the small circle's angle is 1 radian (ie 57°) then the lenath

of the arc o~the small circle is r, it's own radius_

In other words, ~ _ 

length of arc

1 ~ r

Solving this equafion for the length of the ara ~ gives us the equation for arc length, ~ = Y~

P.Xam}~le: Fint3 the length of the aze of a circle

whose radius is 2 meters anal

the central angle is .25 radian (ie 1/• radian).

An.Srver. The arc length ~ rL1B

s = (Zu~eters)(.25)

s = .5 meter or '/~ meter.

ti

I:'xample: Find the length of the arc of a circle

whose radius is 10 meters and

the central angle is S radian (e 1/z zadian}.

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MAT182 Tri~onamehy Page 8 0£ 73 Spring 2410

Area of a sector A= 2 rz8.

.A~ Ehe area of a slice of pie (xe a sector of a male formed by a cenfzal angle) is found

lay ~nulfiiplging 2 tunes Ehe radius squared times the angle an radiarra

Big angle Big area

small angle sma11 area

Suppose the small angle is 2~r . The azea of the circle is ~rr2

B A

2~ ~ m'

Solving this equation for the area 1~ gives us the equation for area, A= Z T2B

Example: Find the area o£the sector with radius 2 feet and an angle of 30 ° .

Answez: A 2 rzB Dzaw the area of the sector.

A ZIlJ~.~61

A _

ZC1JC~~

A = --.square feet .

3

A ~ 1.05 square feet

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MCCCll/Martinez 01250

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MAI182 Trigonometry Page 9 of 73 Sgriag 20I 0

Example: Fiztd the area of the sector with radius 6 feet and an angle of 2 redians_

An~wrr UJT1iP. Y}lP. ~(11TT117IA ATIf~ f~fAW t}IP AT'P.A Ilft}1P CP.("~f)7_

distance d

1]zstance = (rate} (time) d = r U t ~ rate = - ie r = —

time t

~i.izzear sRee~ = v of azi obpect traveIulg um a code is #ocu~d by divicliztg tY~e arc length s

bg the time f: v = f Iength per unik of time like Feet pex secQzui az mites per hour

Tinear speed = v = d f

~d~ ~, rams

Angaiax seed= cu = ~„~~ of ~~e

An~n2ax $peed is hover the hirning rate of an ~~ine is desor~ed. for ins#-ancer an

eng'me zdling at 900 rpm (zevolntions ~Zerminute) is one that rotafes at an angu~ax speed

revolutions 90D revolutions radians radians

of 900 _ ~ =.1800~c

mi~tute minute 1 revolution minule

Suice angular speed, w, is given in terms of revdufions per unit of time,

you must convert ie to radiAns per unit of Time.

There is an important relatianslup betc~tee~ linear speed and angular speed:

iitrear speed= v ~ . S = r • r~ so v = r • ry

t

R~n~mber that lineax seed v = t has the dimensions of length pez unit of time (such

as fee# per second oz Holes pex hour

Inv = r ~ m ~ the radius Qf the amzIar motion r has the same length dunension as s,

ie i~ s is in feet then r is aCso in feet.

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i

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MAT] 82 Trigonometry Pale 10 of 73 stead zaro

_ SLlaf.IIl.aS~ O.f £03alIlIdS

distance d

distance = (zafe) {{isne} d = x U f ~ rate = ie r = —

time t

I8Q degees = n zadians

Arc Yengt~x s = r • 8 = (the lex~gth of the radius)( the sze of the angle in. radians}

Azea o£a sec#c~c A~ z r'B = (? ) (khe radius squared}(tie angle in radians)

s lenoi3raf the arc

Lineax speed v = t = riot of time likefeet per secc~:td os ziules per hoax

Aso, v = r w = (radius of the circular motion)(an gular speed)

D cereirpl angle in radians

Angular speed w = i = tGme~~psed = radians ~ unit of time

~.~ .AL'C ~E*It~EII. ~_T@d. 3~LCI ~P~OC~~ ~OIXkE*S~[1T~C

L denotes the length of the arc of a circle of radius r subtended by the cenfral angle B. Find the

missinS ~Pzftnt~tY rounded to tluee decimal places_

71. r = l0 meters, B = 2 radian, L = T 73. B = 3 radian, L = 2 feet, r = ?

75. r = 5 miles, L = 3 miles, 6 = ? 77. r = 2 inches, 8 = 30', L = 2

In problems 79-85, A denotes the area of the sectoz of a cucJ e of a radius z formed by the ezntcat

angle 8. Find t}ae missing quantity rounded tc~ three decimal places.

79. r = 10 meters, 0 = 2 radian , A— ? 81. B = ~ radian , A = 2 scjuare feet, r = ?

83. r = 5 miles, A = 3 squaw miles, 4 = ? 85. r = 2 inches, 8 = 30', A = ?

87. Draw a ci~le with angle 3 and radius 2 ft. Find tt~e azc length ~ and the area ~1, both rounded

to three decimal places.

88. Draw a circle with angle 6 and radius 4 m Find the arc length L and the area A, both

rounded to ibree decimal places.

89. Draw a circle with angle 70' and radius 12 yds. Find the arc length L and the area A, both

rounded to three decimal places.

90. Draw a circle with angle 50 ° and radius 9 em Find the uc ]ength L and the area A, both•

rounded to three decimal places.

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MA.Tl82 Trigonometry Page 11 of 73 Sprig 2010

~1.4 Right-Trifle `~riganom~fr~ .

Aright triangle Uas the famous PydaagorrBn Theorem refaced tQ it: a~ + b1= cz

(leg)Z+(leg}j = (hypotenuse)

Draw the rectangular axes so the acrWe angles sIwwn below have the vertex of each angle at the

origin. Form two rig(if-triangles by dropping a perpendicular line from the rip of the terminal

side to tha x-axis. The tiriangtes are similar triangles so they have propoction~e sides Label the

sides of the larger iria~vle 9, 12, and 15, and the sides of the smaller triangle 3, 4, and S.

tcttnxnal side

e

~b~ ria~

The trigonometry functions are zatios of the lengths oftwo sides of a fight triangle.

Each ratio depends on the size of the angle B and not the length of the sides ofthe triangle. Note

that in each triangle, the za~o foffieci by the terminal side oven the initial side is 4

The six triganometrie functions are; sin B cos B fan 8 csc B secB cotB

Sang is mad "the sine of theta,"

Yau cannot separafa them. The sin8 is a ratio, However, sin vriYhaut B is a `~iecado;' a sin,

not a ratio. Therefore, always tYrite ifre trig fu~ctiarr with the angle,

sine Hypotenuse

r

~ ~ppo~~ a

The angle is called the arjument of the Cuuctioa fl y

A.dja~cntto 9

x

The ratio called sin B is fozmed by the length of the side opposite the angle oven the lenoath of fhe

side collect the hypotenuse: sin 8= `~

The ratio called cos B is formed by the length of the side adjacent (next to) tl~e angle over tt~e

Xeno h of the side call ed the hypotenuse: cos 4 = ~,"~`~°~~ '~

The ratio called tan B is farmed by tf~e lenb h of the side opposite the angle over the length of .

the side adjacent (new to) the angle: tan 8 = °,,~P°o°~„`

MCCCDlMarfinez00463

MCCCD/Martinez 01259

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 19 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 40 of 122

MAT182 Trigonomehy Paoe 12 of 73 Spring 2010

Whett graphing, the sing is referred to as the `~-coordinate" and the cos9 is referred to as

the "x-coordinate." Therefore, a point has the coordinates (ws 8 , ssn 9) instead of (x, y) .

". O"and " y " are the side apposite the angle. ".~ "and " x "are the side adjacent (new) tq fhe angle.

The hypotenuse " h " is also Down as the radius "r " of the right trian~te.

~acb. ratio has a name that depends on the angle B. The 6 ratios are formed as descnbed below,

_ Opposite

sine of B— —

fidjacent

cosine of B = tangent of 8 =

Opposite

hypotenuse hypotemtse Adjacent

sin8=~=~ cos8=

h=r ~8^O=x

_ hypotenuse

cosecant of D— — secant of B = 

hypatem~se cotangent of B=

Adjacent

Opposite Adjacetrt Opposite

h r

cscB=O=—

h r

sect=~=s

g

cot6=O=—

x

Y Y

Nofice that the reciprocal of the sine frinction is the cosecant function. Tha reciprocal idenSities

aze provided below.

Reciprocal Identi#ies csc B = 1 sin B = ~ sec B = 1 cos B = ~

sin8 csc8 .cos8 sec8

cot B = 

1 = cos e tarp 8 = ~ = s~ B Memnrize these reciprocal identities_

tan B sin B cot B cos B

Tt',Xample 1 Find the value of the si t higonometric functions of the angle B in the triangle.

We see that the hypotenuse =~ and the azljacent side is = 3.

7'o find the length of the appo,s~ste side, we use the Pythagorean 2~reorem. 5

a2 -f- bs _ ~i op~nsite 8

(leg)Z + (leg) = (hypotenuse)Z p

(adjacent)z+~opposite~~ = ~hypotemuse~Z 3

3z -i- ~opposite~i = SZ

(opposite) ~= 25 - 9 =16

opposite = 4

Now that we know The lengths of the threw sides, we can find the six higonometxic functions:

ot~ a

sing = °~°~ - ~ cas9 = 

edjQce:d 3 ~g edjncent 3

byjuteanse 5 6y]utenuse ~ 5

hy~vtetmsa 5 hy~ten~ce 5 'adjerenf 3

acB= opmsffe ^ 4 ~c8= adJ'nrtat 3 cot@=opFositrxd

MCCCD/Markinez00464

MCCCD/Martinez 01260

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Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 41 of 122

MAT18Z Tri~ouometry Pa~c ]3 of 73 Spring 201D

EXAMP~.E 2 sin B = 5 Find the rem~iriing 5 trigonometric fimctions of 8.

SoInfion_ cos 8= Label the sides of a right triangle:

SlIl e 5 _ 1

G3A19 = "` ~ 2 ~ 2

COS B 5

a~ + bz — c~

CSce = sina Sece ~ cosB

cots =may =

Steps for T~uxdin~ the Valnes of 5 'I'rigonometcic unctions When One is Known

Step I : Draw a ri ght triangle showing the arxgle B and label two sides of the triangle.

Step 2: Find the length of the third side by using tine Pythagorean Theorem.

Step 3: Use the definitions to find the value of the remaining 5 trigonomehic fimctions_

Example 4 sin B = 3 . Find the enact value of the rema~n~ng five tirigonametric functions.

Solution: 1) l?raw anri label t}ze sides of a richt triangle va Quadrant 1.

2) Use the Pythabwrean Theorem to find the adjacent side,

~z ~ bi _ ~z

. (adjacent side)= +(opposite side)' =(hypotenuse)

(adjacent) ~ + 1 ~ = 3 Z .

(adjacent)Z + 1 = 9 1

(adjacent)Z = 8 Q

adjacent = 2f

3) Use the definitions to find the value of the remaining five trigonomehic functions.

az£acad opposite _

cos B = hypotenuse _ ~ e _ a~.aani —

cSc B = 

b~°~nue -

opposite

z3 4 3 '4 2~

sect = 

otcnasc ^ cDtB 

=adjacent

adjacent opposite T

MCCCD1Martinez00465

MCCCD/Martinez 01261

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Case 2:1.2-cv-00702-DGC Document 68-9 Filed 1.0/11/13 Page 42 of 122

MATI82 Trigonometry Pa~~ 14 of 73 Spriug2010

Example 5 If tan B = ~ ,find the exact value of the remaining five trig fimctions_

1) Draw the right triangle where tan B ~ ~ =

opposite a~,o~,r I , o~~

2 adjacent

z- ey,Wt

2) Use the Pythagorean Theorem to find the hypotenuse.

3}Now use the definitions to find the value of the other S trigonometric functions.

siA B = hypotcnvse — 

cos 8 ~= hypotenuse —

ny~otm~~ hypot~,~~ ~~~

csc 8 = adjzcent — 

sec 0 = ~~aceut ~ 

cot B = opF~~t~ _

~.4 R~ht Triangle 7~ni~auQmetr,~ Hc~n~tee~ark

Find the value of the six tzigonometric fwictions in triangles 11-19.

il. t3. B 15 8 4

5 2

' 3

12

1

17. ~ ~ 19..

1 .(_''

—_ e B

i

Foz X21 and 23, use identities to find the value of the rema;n;,,g trigonomehic functions .

21. sin ~ = 1 , cos B = ~ 23. sin 9 = 2 , cos B =

2. 2 3 3

In problems 25-35, find the value of the reiziaiuing five trigonometric ~mctions.

25. sin ~ = ~ 27.. cos B = ~ 29. tau B = I 31. sec B = 3 33. tan 8 = f 3 S. csc B = 2

2 3 2

MCCCD/Martinez00466

MCCCD/M~►•tinez 01262

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Case 2:12-cv-007Q2-DGC Document 68-9 Filed 10/12/13 Page 43 of 122

7vfATt82 Trigonometry Pagc 15of T Spring2bl0

1 5 T'he Pvtha~orean Identity and Comp7etnentary Functzons

Pythagorean Identities

You now ]mow eAough so we cau derive the ppthagorean Identities_

The Pythagorean Theorem states f3iat LZz -f- bZ — CZ, which we can wrzte as

(oFpos~te) 2+ (adjacent) 2 - (hYPofenuse) 2

Dividing each side by the (hypotenuse} Z we get

c

a

b

~Siu8~2 +~cos8}2 = I

"~'he Pyt}~agozean Identity S]St2 8 -i- GOSz 8 = 1

bivide the Pythagorean Identity by cost 6 and simplify.

5in~ 8 + cosz B =1

.Divide the Pytl~sgorean Tdentrly by szn' B and simplify.

~; S1IlZ e-l- COSz B=~ .

h

'~`

`'' PY'C~'iAGOREAN ZDEN~'~'I'l1L`~S Commit these Pythagorean Xdentities to memory.

~~'! sinz 6-i-cos29=1 tanz B-1~1.=secZ B cotz B+l=csc~8

t~.

MCCCD/Martinez00467

MCCCD/Martinez 01263

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 23 of 243
Case 2:12~cv-00702-DGC Document 68-9 Filed 10/11/13 Page 44 of 122

,v .:

MAT182 Tzi~onometxy Paga 16 of 73 Sprittg 2010

IZecaI1 that two acute angles are called cotnplemenfaty if ttiep add np to 90 degrees. Also, the

three angles of any triangle add up tc7 1$0° , So in a right triangle, the twa acute angtes aze

complementary angles_ Additionallq:

The sine and cosine aze called cofuncfions. ~

The cosecant and secant are cofimctions. c ~i ~r ~ ~ Opposite of/j

'the tangent and cotangent are cohmctions_ ~ A

• Adj amt to ~

Whic•.Ii leads us to ~ 

opposite ❑fcc

The Complemenfary Angle Theorem: Cofnnciions of cozn~Iementary angles are equal.

sin /j = 

°t~Pasife _ cos a 

Comptemrnraxg ewes"

hypoternue ~..«--~~

si~3o= = ~ossa°

adjacenf ,~ ,~

cos~3= =sing

hypotenuse Co~~~~,~

- ~~; °.PPosite =cotes 

~omFiane~a~yarn~ts .

adjacent ,~--------~,

III ~~r = C0~ ~ Q~

hypotemise ~„ ,~

Esc ,Q = =sec a

opposite C ~~

hypofenuse ~ nmpirmen[ary axc~les

sec/3=— =cs~a

acl~'acenf .~""—~"3~

S~C$Q+= CSC~~~

cot/3 ~ 

adjacent _ tan a . ~` ~F

opposite C afimcfiaat

Because of these relationships, the sine and oasine, the tangent aad cotangent, and.

the secant and ensecant are called eofunefions.

i:zample 6 Using the Complementary Angle Theorem, find the corresponding cofunctions.

s:n 62' = ccs (90° — 62~) = cos ~ tan 2 =cat

cas 4 =sin _~ nse 6 =sec

MCCCD/Martinez00468

1VICCCD/Nt~r•tinez 01264

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Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 45 of 122

18 T'~ o ~o of S ~ n20 0 MAT 2 n5on merry Pam 17 73 prtn~ 1

Eaample,7 Find tl~e value of each expzessioa Do not.use a calculator.

(a) sec 28" - csc 62' (b} sin 35°

cos SS°

1.5 The Pythagorean Ydentity and Complementar~Funcfions Homework

In Probieu~s 37-53, find t~ze exact•value of each expzessian. Do not use a calculator_

37. sin'20'+cosZ20' ~39. sin80'—cos52' 41, tan50'-- LSD 43. sin38'—cos S2'

cos 50'

~5. cos10' 47. I—cos'20'—cos270' 49. tan 

20'—C0s70'

sin 80' cos 2d'

S1. tan35'•sec55'•cos35' S3. cos35'sin55'+cos55'san35'

55_ Given sin 30° = ~ , use trigonometric identities to find the eacact value of

(a) cos b0' (b~ cos'30' (c) csc 6 (d) sec 3

57_ Given tan B = 4, use tcigonomelric identities to f nd the exact value of

(a) sec ~ B (b) cot B (c) cot (~ — B1 (d) csc~ e

\2

59. Given csc D = 4, nse trigonametri.c identities to find the exact value of

(a) sin B (b} cote B ~(e) sec (90' — B) (~ sect B

MCCCD/Martinez00469

MCCCD/Martinez 01265

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 25 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 46 of 122

MATI82 Trigonometry Pagc 18 of 73 Sprin~2oto

16 Compufin~ fhe Values of Trig; Functions for 5vecial Trian~Tes

The special antes in trigonometry aze 0° , 30 ° , 45 °; 60 ° , 90 °, 180 ~ , Z74 ° , 36fl

3x ~ 27c

In zadian meastue these an~PS ~C Q , b q a 3 s Z ~ ~ ~ 4

cos 90°,sin 90~

(0 , 1)

~cos180 ,sin180~~

(—I , 0)

0

~cos0 ,sm 0

{I , 0)

~cos27o°,sm270°}

~~ > 1)

Commit to memory tlae values of the 6 trigonometric functions

for angles 0 ° , 90 ° , 180 ° , 27Q ° , 360 °

s

s3n0~=D szn90°= 1 sinl$Q°=0 sin270~ = —1 sin36~° = 0

cos0°=1 cos90°= D cos 180°=-1 cos 274° = 0 cos 360° = 1

csc 0 ° = undef csc 90 ° = 1 csc 180 °= undef csc 270 ° _ —1 csc 360 ° = undef

S~ p ~ - .~ ~c q0 ° = undef sec 180 ° = -I sec 270 "= uudef sec 3b0 ° = 1

tan0°=0 tan 90°= ,mdef tan180°=0 tan 270° = unacE tan 360° = 0

cot 0 °= undef not 90 ° = 0 cot 180 °=' undef cot 270 ° = 0 cot 360 ° =undo£

MCCCD(ManinezOD470

MCCCD/M~rtinez 01266

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 26 of 243
Case 2:12-cv-007Q2-DGC Document 68-9 Filed 10/11/13 Page 47 of 122

MAT182 Trigonometry Pagc 19 of 73 SFxin~2036

Consider an equilateral triangle cut in ha}f so the top 66 ° angle makes two 30 °- angles.

Each ri ft triangle is now a special triangle known as a 30° -60° - 90° triangle.

~~

o" 3a''. .

n• z ~ ~

2 ~ ~t ~ 2

6D+ ta~`~

The sides a~the right #'angles rrzeasure I, 2, ~. The h~potenusa ts~eas~zres 2.

the side opposite the 30 ° aaeasures i, and the side opposite the 60 ° angle measures ~ .

Commit to memory the trigonometric function values of the two very special triangles

often used in trigonometry. They are the

45° 45° 90° or ~ , '~ , '—~ and the 3 0° 60° 40° or ~~ , ~~ , '~

2 2 Z 

6 3 2

z ~. 'o

~ Z ~ ~ 6D 1

its ~ '' ~o' ~ .

sin 45 ~ = ~ ~ csc 45° _ ~ = z 2

cos 45 ° = Z sec45° _ ~ -- 22

fan 45 ° = ~ ~ 1 cot 45° = ~ = 1

szn30°= 1 sin 60°=

2 z

~s3o°— ~ ~s6o~= 1

2 2

tan30° = ~ = 3 tan 60° = f

MCCCD(Ma r`iinez00471

MCCCD/Ma~~tinez 01267

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 27 of 243
Case 2:12~cv-00702-DGC Document 68-9 Filed 10/11/13 Page 48 of 222

4T182 Trigonomcti}' Yee 2Q of ?3 Spring 20f0

;ample 1 Find the valszes of the six k~zgonomeiz3c functions of 4 = 45~_

~tu6on Draw the triangle and label the sides and the angles.

7C p 7t

suz—=sin 45 = cos--~

as' ~ ~ 4 .

2

n ~c

45~ SBC' — _CSC^ =

4 4

'•

~ n

tan,._.. - cot

4 4

ample 2 find the exact value of each escpression.

(sin 45')(tan45') • b) sec 4 ~ cot 4

z~; ~'_.

:~:t=~

Viz::

ARCCCDRvtartin ez00472

MCCCD/Martinez 0]268

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 28 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 49 of 122

r~'+~'

fti%

s:_

tiE~:

~'~

+:d; :::•

,,. ;.

q,r_

"'r

1

::

i~~~

pl:

%:

~_:•.>

MAT182 Triaonomehy Page 21 of 73 Spring 7010

~zample 3 Find the Exact'Values of the six hiaonometric functions of 6 = 30' and 3 = 60'

Solution: Form a right triangle and label the sides and the an;l es appzopziately.

~c 1t

szn 6 = cos 3 =

2 p' _

3 n

cos— = szn— _

6 3

60~ .

{ate ~ = cot ~ _

1 6 3

~ ~

csc 6 ~ 5~ 3

~r ' n

sec b = csc 3—

n n

cot6= tan3= .

+~ratnple 4 Find the exact value of each expression.

(a) sin 45' cos 30' (b) tan 4 -~ sin 3 (c) tang 6 +sin'

~xaimple 5 Use a calculator to find the approximate value (to two decimal places) o£

• (a) cos 48' ' (b) csc 21' (c) tan i2

Solution Since these are not the special angles, we will use the calcu}ator as follows.

(a) First, we set the MODE on the calculator to receive degrees.

(b) Most calculators do not have arse key. However, the csc 21' is 1

cos 2I°

(c) Set the MODE on the calculator to receive radians.

MCCCDtMartinez00473

MCCCD/Martinez 01269

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 29 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11!13 Page 50 of 122

,.--

s->:.~

c~.'r

rr=~:~

'YLi i:_.~.

'c

~:'

':

'za'

} '~

z::

';+>

~~;s:

~~':

=zx

IrfAT182 Trigonometry Pale 22 of 73 Spring 2010

1.6 Computin6 the Values of Tri~Frxnctiorts for Special `I'rian~les IIomework

Jn problems 7-Y 6, f {~ = sin B ,and g(B) = cos B . find the value of each expression if B = 60° .

Do not use a calculator.

7- .f(B) 9. f ~~ + ll.~f ~9~~2 13. 2 f(8) 15. f 2B~

In problems 17-27, find the enact value of each expzession. Do not use calculator.

17.4 cos .45' - 2.sin 45' 19.6 tan 45' - $cos 6U' 21. sec 4 +2 csc 3

23.sec2~-4 25. sin230'+cosZ bO' 27. 1.—cosz30'—cos1 60'

6

In problems 29-45, vse a calculator to find the apprmcimate value of each egression. hound the,

answer to two decimal places.

29. s'va 28' 3I. tan 21' 33. sec 41' 35. sin n 37. fan ~5

10 1Z

39. sec ~ 41. sin 1 43. sin!' 45. tan 03

12

47. Applications and Extensions Find the range R and maximum. height H rounded to 2 decimal

places.

Projectile Motion The path of a projectile 6sed at an inclinaticin Bto the horizontal with initial.

speed uo is a parabola (see the ftgure). Tie range ~t of the projectile, that is, the horizontal

distance that the proj ectile travels, is found using the fzmcfion

~~

B\ _ ~.Up 5ll1BCOSB

I g

Where g ~ 322 feet per second ~ 9.8 meters per second is the acceleration due to gravity_ The

maxi.muna height.Ko£ the pzvjectile is given by the fimction

~p gjn~ B vo w Initialspeed

. Ht~)—~ 2g t

' Neipht.7f ~.~

. - _ ?r.8' ^.FAQ:μ.;+:. -<.rS~, ~.,s. ~ _ .. .u.

f~ange:R— ~~

Find the range R and tttaxim~ height Ii to two decimal places when the projectile is fired at an

angle of 45 ° to the horizontal with an initial. seed of 100 feet per second.

MCCCDIMartinez00474

MCCCD/Martinez 01270

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Case 2:12-cv-007Q2-DGC Document 68-9 Filed 10/11/13 Page 51 of 122

MA.TI&2 Tri;onomeiry Page 2; of 73 Spring 2010

1.7 Tri„oaometric Fnncfions of General Au~les.

,Example 1 Find the value of the six tzigonomctric. functions of a positive angle 8 if (4, -3) is a

point OA its terminal side. EGnt Start by diawmg the angle and the coacspondin~ Mangle. Label the ricks.

Ex~.mple Z Find the values of the six h-igouometzic functions of

T (a) B = 0 = 0° (b). B = z = 90” (c) B = ~t =18 0' (d) B = ~z = 270'

Answers: .

a) sin 0=sin 0~ = 0 b) sin 2 = 1 c) sin n =.0 d) sin 2 ~ -1

cos 0 = 1 cos 2 = 0 cos ~t = - I cos 2 = 0

tan 0 = 0 csc 2 =1 tan n = 0 csc 2 = -1

sec 0 = 1 cot 2 ~0 sec ~r = -1 cc~t 2 = 0

23 ese 0 &cot 0 tan z and sec Z csc.z & cot,z tan &sec 2

are not defined aze not defined are not defined are n.ot defined

MCCCD/Martinez00475

MCCCD/Martinez 01271

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 31 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 52 of 122

i,+'

,~•'f~.

R~~i

~'

~:

4': ~:

~•}

:=n

-,;

MA.7182 Trigonomehy Page 24 of 73 Sgriag ZO10

Example 3 a) sin390 " b) cos 420 ` c) tan 4 4 c~ sec— 4) e) csc(-270°)

Hint It is best to sicefch the angle first

a) The angle is coterminaI with 30 " .

sin390° = sin(360° +30°) =sin3D° = z

b) c) d)

~xampla 4 Name the Quadrant in Which the .Angle Lies

(a) sing<0 and cosB<0 (b) sinB>0 and cosB<0

(c) ~ ~d)

y

B

Bder~~ mgl< A`~

-- %

e)

MCCCDIMartinez00476

MCCCD/M~rtiuez 01272

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 32 of 243
Case 2:12-cv-00702-DGC Document 68-9 Fifed 10/11/13 Page 53 of 122

MATI82 Trigonamehy Page 25 of 73 Sprin~2~10

Example 5: Find the sine of the reference angle for each of the following angles:

(a}150' (b)-45' (c} ~ (c~— ~ _

Solution (a) sin 30' = z {b)

y

ry isa

3~~ z

T (c) (d)

Steps for finding the values of the trigonometric functions of an Angle

Step 1: Tf the angle B is on the x-axis or y-axis, label a point on its terming] sidq and name the trig values.

Step 2: If the sngLe B is in a quadrant, draw a triangle and label its sides, then name the trig values.

Example 6: Find the,value of the following trigonometric functions.

17n n

(a) sin 135' (b) cos 600' (c) cos 

6 ~~ ~ ~--3~

~ampie 7: Given that cos B = — ~ ,'~ c B < n, find the value of the remaining frig functions.

3 2

MCGCD/Martinez00477

MCCCD/Martinez 01273

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 33 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 54 of 122

^''i3'~': ,i:ts:

':~

=e;_'

;. .;:

ir''

+:;e

r ::-:fir;:" >~:

:4~~~-

MAT182 Trigonomehy Pa;e ?b of 73 Spring20T0

Example 8. If tau 8 = -4 and sin B < 0, find the value of the zemaining trigonometric fvncflons.

1.7 Tri~onornetric Functions of General A.n~les planie~vork

In problems I1-20, a point on the teAninal side of an angle B is given.. Puad the valae of the six

higonometrsc functions.

11.~--3,4~ 13.(2,-3) 15.~-3,-3~ 17.~ 2 ,~

l J

Por #21-32, find the value of each expressioza. Do not use a calculatoz.

21. sin 405' 23. tan 405' 25. csc 450' 27_ cot 390' 29. cas 

34

In Problems 33-39, name the quadzant in which the angle B lies.

19.

2 ,- z

31., tan(2J~r)

33.sinB>O,cosB<D 35.sin8<O,tau9<0 37.cosB>O,cotB<0 39_seoB<O,taaB>0

In Problems 4I-57, name the ieferenee angle of each angle. 4I. -30` 43.120'

45. 210' 47. ~ S 49. 3 g 51. 135' S3.-- ~ SS. 440' S7. 4~ '~

Por #59-88, find the value. Do not use the calculator. 59. sin1S0' 61. cos315' 63. sin510'

65. cos-45'~ 67. sec240' 69.cot330' 71.cos ~ 73. cot b 75, 

cosl4

77. sin(- 23 1 79. tan 3~ 1 81. csc~-315' 83. sin~S~r~ . 85. tan ~7~t} 87, sec-3~r}

~ J

In Problems 89-106, ftud the value of the remaiu.uzg tzigonomeYric functions.

89. sia 8 =13 ' 8 in Quadrant II 9l . c;os B = - 5 , Bin Qnacirant III

33. sing=3, 1 90'<B<180' 95. cos8=-3,180'<B<270'

97. sing-2; tanB<0 99. sect=2, sing<0 101. tang=3, sing<0

3 

4

103. tang=-3, sinB>0 105. cscB=-2, tanB>0

107. FindtheexactvaIueof sin 45'+sin235'+sin 225'+sin315'

1~9.~f~e~=5~e~o_2, rya fro+n~ k1i.z~~te~=r~a~~,findF(B-rz)-

113. If sin8= 5, ~a~s~e.

MCCC~R~Iart'nez0047S

MCCCD/Martinez 01274

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 34 of 243
Case 2;12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 55 of 122

MAT182 Trinonnmctry Pa~c 27 of 73 Spring 2010

1_S ~+rcle Tri~nnamet~-~

Let t be the length of the arc fzom the xsaxis to the point 1' _ ~~ , 1~ on a circle.

/`~~_~~

r .~'

B = t radi~ocs

sing=sint=2

~ cosB=cost= 2

Example 1 1' _ ~-1, ~) Find the value of tt~e corresponding 6 #rigonomeinc fw~ctionsy'~ Solution: sin t =

_f~

The unit circle has a radius of 1.

x

Domain & Renge of Trigonometric Funcrions

Function Domain is fhe angle

f (B) = sin D D =All real uvmbers

f (B) = cos B 8 =All real numbers

f(B)=tanB 9 = (~,-+w), esceptoddintegezmultiplesof Z

f (8) = csc 8 8 = (- <o, +m) ,except integer multiples of n

f (D) =sec 8 B = (—~, +Lo) , except odd infeger multiples o£ Z

f (B) = cot B B = (-moo, -~ ~) , except integer multiples of ~r

Range is the function

sing = [-1 to 1~

cosh = [-1 to 1]

tan 8 = (—~o, +w)

cscB <_ —1, csc8 z 1

see8 5 —1; secfl ? 1

cot 8 = (--~, -1 a~)

MCCCDlMartinez00479

MCCCD/Martinez 01275

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 35 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11(13 Page 56 of 122

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.i

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MATI82 Trigonomctry Page 28 of 73

Periodic funcEions repeak after a certain time.

Z7te frignrometr~ie, functions repeat so r}zey vre periodic functions.

2n is tFze repeating perzod far' sine, cusiex~ cosecarzE, and secant.

sin{B+2~r}=sinB cos(8-r2~c)=cosh

csc(B +2n) = csc B sec(B -1- 2x) = sec B

s~rin~ Zoio

Far Tangent r# is ~r.

txn(B + n} =tau 9

cot(B+n) =cot B

Exa~wpFe 2 Rind the exact value of a) sin420~ b) tan Q S c) cos Q~1 

Even. fancfions aze symmefric about the y-axis odd fiutcfzons wont ~1~.e origin„

Reca}I tYiat a function is evezz if f (-x) = f (x) 'The sign of the x changes but'the y doesd f.

A funcfian is odd if - f (-x} = f Cz) 'I~e sign of x and y changes, but the graph does~i t,

cos~6~ ~ sin~6~

MCCCD/MartinezD0480

MCCCD/Martinez 0]276

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 36 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 57 of 122

` MAT182 Trigonocnetcy - Pale 29 of 73 Spring 2010

The casino ai~,d Ehe secant are even £unctions: cos(--8) = cos B sac(—~ =sec 8

The rest are odd functions: sin(—D) =—sin B tan(—B) = —tan B

csc(- $) = —csc B cot(—~ _ —cot B

Exampke 3 Find the exact value of: a) sin(-45°) b) cos(—n} e) cot— ~ ~ d) tan ~— ~~~ 3 

Z_8 ~rcle `I'riponnmef-~-~ IIozn~ewo.~-k

For #?_1-36 find the exact value. Do not use a calculator_ 21. sin 405' 23. tan405'

25. cos 450' ?7. cot390' 29. cos ~ 3 31, tan~2l~r~ 33. sec ~ 1 3 S. tan 6'~ ~ 

Far #37-54, find the exact value of each eacpression- Do not use a calculator.

37. sin-60'~ 39. tan-30'~ 41. sec--60'~ 43. sint-90'~

45. tan(-4~ 47. cos—~J 49. tan—~r~ S1, cscJ-4~ 53. sec -6I

~ \ l

In Problems SS-G0, find the exact value of each expression. Do not use a calculator.

55. sin —~r~+cos~5~r~ 57. sec —~r~-I-cscl —~l 59. sing —~~~—tan —~~9

l 2.1 ` 4 4

61. What is the domain of the sine fimction?

63. For what numbers B is f ~fJ~ = tan B not defined?

65, Por whai numbers B is f ~B~ = sec B not defined?

67. What is the range of the sine function?

69. What is the range of the tangent function?

71. What is the range of the secant function?

73. Is the sine function even, odd, oz neither? Is its graph symmetric? With respect to what?

75. Is the tangent fimction even, odd, or neithez? Is its graph symmetric? With respect to what?

77. Ts the secant function even, odd, oz neither? Is its graph symmetric? With respect to what?

79. IfsinQ=0.3,findthevalueof: sing+sin~0+2~r~+sin~B+4n~.

S1. If tang=3, find thevalneof tan8+tan~9tzr~+tan~B+2zr~.

MCCCD/Marfinez60481

MCCCD/M~rtinez 01277

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 37 of 243
r<'r~

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7'~ '`;

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. s+::.

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Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 58 of 122

MAT182 Triwnometry Pa~c 30 of 73

~_4 CYraphs &Transformations o£thc Sine and ~asine

Graph of Yee S9ne Function

y~sinx, —w<x5-i~

y=sin z, 0_<x52n angle

Y ~ O

v=sw s, OS.ES?b

t z 1~\~0

6'3

r ~ 3x

t t'6 ~~'~ ' 116 `_'~

}2 '-1~

Spz-ina 2C t0

sin x

MCCGDIMartinez00482

MCCCD/Martinez 01278

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 38 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/x.1/13 Page 59 of 122

5r :'.

MAT182 Trisonomctry Pa~,e 31 of T

Graptu of the Cosine Function

y=cosx,0_<x<_2~r

~~

i

N.» f~.~ ~

~. x

T ~ Y

(~, -t)

Spring 2010

angle `Y' cos z

MCCCD/Martinez00483

1VICCCD/Martinez 01279

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 39 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 60 of 122

~..

':><'=i

MATl82 Trigonomehy Page 32 of 73 Sptuig 201 D

Horizontal shiffmg of fhe graph.

EX,ANfpT,E X Sine sltifling Use the graph ofy = sinz to graph y = sin ~x— ~ J

.

Ezample 3 Use t3ie Gzaph of y = cos x to graph y = 2 cos x

MCGCDIMartinezD0484

MCCCD/Martinez 01280

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 40 of 243
=`'-'<

~h,..

F=:..

k`'~

~"::

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;' z

Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/1.3 Page 61 of 122

MATI82 Trigonometzy Pace 33 of 73 S~zmg 2010

"1" i.s the amplitude C. e, the largest vertical value} of the sine anal cosine functions.

•` 2~r " is the period of the sine and ursine fcutctions.

E~cample Use the Graph ofy = cos x to graph y - 2 cosx .

In example 3 we obtaiz~eri the drap6. of y = Z cos x ,which is reproduced below. Notice that the

y-values of y = 2 cos x lia between —2 and 2, inclusive.

• v

y = 2rosx

c

Theorem If ru > 0, the azuplitude and peziod of

y = Asin(~x~ and y = AGOS~GJx~ aze given by Amplihule = IAI

Sine &Cosine Period = ~2

m

Y = A sin( a~x — g~ )

~ is the phase shy ',4I is amplitude mx is the period

example 5 Determine the amplitude and period of y = 3 sin(4x).

27C ZT 7t

Solviion: Amplitude = ~~ = 3 and the Period = ~ = A = Z

x

MCCCD/Martinez004B5

MCCCll/Martinez 01281

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 41 of 243
Case 2:22-cv-40702-DGC Document 68-9 Fi{ed 10/11/13 Page 62 of 122

A

MAT182 Trigonometry Page ~4 of 73 Spring 2D 10

Example 6 Graph y = 3sin(4x)

Is.~l Cr.~7

~ ~~

f4Y ~€~.D)

- -4 f 'f ~r r

Solydion: -~

t-:.at c~.-+~

Example 7 Determine the amplitude and period of y = —4 cos(~rx) ,and graph the function.

aw ' n~

~i 

-~ ~ , i

Y = rL sin( l,~x — g~) 9' is thephase shift I

'A) is amplitude r,~x is the period ~~~ '~

~ ~ Graphs &Transformations of the Sine and cosine Homework

In Problems 21-35, use transformations to graph each function.

2I. y=3sinx 23. y=—cosx 2.5. y=sinx-1 27. y=sin(x—~r)

29. y=sin~.zx~ 31. y=2sinz+2 33. y=4cos~Zx~ 35. y~-2sinz+2

,. Ln Problems 37-45, determine the amplitude and period of each function without graphing.

"` 37, y=2sinx 39. y=--4cost2x~ 41. y=6sin~nx~ 43, y=—~cos~~x~

,~ 45. y=3sin~—~x J

In Problems 61-69, graph each sinusoidal fimction.

61. y=5sin(4x~ 63. y=5,cos(rrz} 65. y=-2cos(2nx~+1

67. y = -4 sin (2 xl 69. y = 2 sin(- 3 xl

l J \ ~

MCCCD/Martinez00486

MCCCll/Martinez 01282

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 42 of 243
Case 2:1.2-cv-00702-DGC Document 68-9 Filed 10/Z1/13 Page 63 of 122

MATI82 Trigonometry Page 35 of 73 Spring 2010

X.10 GraphintLthe T_ aunt, Cotangent, Cosecant, Secant

Graph y=tan x, - 3 < x < 3

x y = tan x

~-,~; ~--1.73

(x . v

~- 3 ,-1.73} - 3

-4 -1 ~-4,~i1

1

6

-- 3 ~-0.58 (- 6 ,~_58)

0 0 ~D,O~

6

3 0.58 (6, _58)

4 ~ t4'1~

3 ,~3 X1.73 3,1.73)

y

y=ranz, -2<x<2 ~3 _ r

.

t3~~~

~ - f4, j)

,~3 ~~.

3 6' 3

rrr ~rr ,r `~~ ~~ ~ 7r X

2 T3 v 6 `~ 6 3 2

6 3 '~

a ~) ~j

3

Aq CCCD/Mertin ez00487

MCCCD/Martinez 01283

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 43 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 64 of 122

M?,T782 T'risonometry Page 36 of 73 Spring 2010

Graph y = tan x, x - (-~,+~o)

r 3z Sn 7~ ~~"'~1i

X= Z~ 2 r 2~ Z''' 2

The tangent function is discontinuous @ ~ 3s _~ _ ~~ _~~,_~~,~

x~_~,_2, z, z> > ~_

natural # n = 1,2>3,... 2n = an even number 2n-1 = an odd number

y =cot x

Domain -

Rau;e =

MGCCD/Martinez004$8

MCCCD/Martinez 01284

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 44 of 243
~;:

r,s?

Case 2;12-cv-007p2-DGC Document 68-9 Filed 10/11/13 Page 65 of 122

MAT182 xrigauomctry

y=cscz

~QI➢317t -

Range =

y=secx

Pa e 37 of 73

x>-2~ x=-~r x. ~ ~ y

i

r ~

1

~ ~ 7

i

1

s

~ 3 ~

1 Z ~ 1

~ , -t

~~ ~

V~ i

t~ i

Domain = age

Sprm~ 2010

x=n x~_;

y = csc .c t =~

(z. l) ~

:4 ..~'.

i r

i '. ii

~ - ,r

{~ 'C ...~.

1 T :4

~ i ~'

~

~ ~'_ , _

~ .-.~

~ 1 _ ~3.

;_

i ~.

i, ~~. is

~.IQ Graphiu~ the Tan~enf, Cotangent, Cosecant, Secant

3, The gaphof y = ~1x is symmetric withrespectto dae _mod has vertical asymptotes at

5. It is easiest to graph y = secx bq fizst sketc(iing the gzaph of

7. What is the y -intercept of y = tanx ?

9. What is the y-intercept of y = secx?

11. For what x-values,-2~r c x < 2n, does the secx =1? For wl~zi valves of x does secz =.-I ?

13. For what x -values, -2~c <_ x <_ 2n ,does the ~aph of y = sec x have vertical asymptotes?

1 S. For what x -values, -2n <_ x _< 2.z, does the ~-aph of y =tan x have vertical asymptotes?

In Problems 2k-44, gaph each function.

21. y=-seex 23. y=sec(x-2~ 25. y=tan~z-n~ 27. y=3tan~2z~

l

29. y=sec~7~~ 31. y=cot~,-cx~ 33. y=-3fan~4x~ 35. y=2sec~2x~

39. y=~cot~z-4~ 41. y=taaz+2 43. y=sec+x+2~-1

l

MCCC~/Martinez004B9

MCCCll/Martinez 01285

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 45 of 243
r;i

st

.:~,'S

a':

Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 66 of 122

M~LT782 7ri~onometry Page 38 of 73 Spring 2010

1.I1 The Laverne Trig Functions

T1ae Inverse Sine, Cosine, and Tangent Functions

When the angle is lmnwn, you can find the value of the trig function.

Find the sine ratio when the angle is 3 : ,

. ~ .

sin.-

3

When the angle is not laxown, you must do the inverse.

Find the angle when the sine ratio is 2 .

sin B = ~ B is the angle whose sine ratio is ~ .

In math symbols tbis looks like 8 = arotan ~ or B = sin ~

Example 1 Find the value of: san ' 1

MCCCb/Martinez0049p

MCCCD/Martinez 01286

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 46 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 67 of 122

MAT182 Trigonometry Page 39 of 73 Springy 2010

~',xample 2 Find the value of a) sin~l ~— 2~ b} cos ~ 0 c) cos' i —

~ J

f 3n

Solution a) 53Il ~ ~-z~=— 6 b) cos 10= ~ c) cos`' — 2 ~

l

Example 3 Find an appzo~ate value of (a) sin ' 3 (b) sin ' !— ~~

Express the answer in radians rounded to two decimal places. Use a calculator.

Solufion (a) sine 3 = 0.34 (b) siri ~ I -- 4 I ~ —025

~ l

MCCCD/Martinez0~491

MCCCD/Martinez 01287

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 47 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/3 Page 68 of 122

Ivi/iTl%2 Trigonometry Page 40 of 73 SprSng 2010

r

- _ Example 6 Find the exact value o£ (a) cos ' f cost 2~~ 1 (6) cosCcos' ~--0.4}~

L. l 1

Solution (a) cos 1 ~cos~IZ !~ = i~ (b) cos~cos ~ ~~.4~~ =-0.4

l

Hxample 7 Find the exact value of: a} tan`' J. b) fian~'

Example 9 Solve the equation 3 sin ' x = n

'I`he solution set is {Z~

MCGCD(Martinez00492

MCCCD/Martinez 01288

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 48 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 69 of 122

MAT182 Tri~000metry Page 41 of 73 Spring 2010

Graphs of Inverse Trig Functions

The(nvecseSineFunction y=siri'x, x=[-1,+1] and 8=y=[—z,-~-2]

B=x-siny

' f (f(x)) =sin-` (sin x) = x when x = C— 2 >+ Z ]

f (f -' (x)) = s'tn(sin ' z) = x when x = [-1. +1]

l'he Inverse Cosine Function y=cos' x, x=[-1,+1], B = y=[ 0, n

8—x=cosy

f -~ (f (x)) = cos ~ (cos x) = s when z = [O,,z]

f (f -1 (x)) = cos(cos' - x) = x when x = [—I,+1]

MCCCD/Martinez00493

MCCCD/M~u•tinez 01289

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 49 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 20/11/13 Page 70 of 122

MAT182 Trigonometry Page 42 of 73 Spring 2010

ThelnverseTangentFunction y=tan-'x, x=(—~,-tzo}, B=y=(-2,+Z)

B=x=zany

f~(f~x)) = tan~~tanx)-z whenx=L-2,~+-2~

f (f ~ (x)) = tan(tan ' x) = x when x = (—w,-8 oo)

f-t(f(x)) =sin 1(sinx)=x whenx=[—,2,+2}

f[f-~(x)) = sin(sin'x)=x wbenX=[-1,+1]

' f (f (x)) = cos ~ (cos x) = x when x = [0, ~r]

f(f`1(x)) = cos(cos t x)-x whenx=[-1,+1~

f'(f(x)) =tan~(tanx)=x whenx=[—z,-F2]

f(f'i(x)) = fan(tan'x)=x whens=(—o',-t-~)

'I'Iie Inverse Cosecant Funefinxt y = csc 1 x, x 5 —1 or z> +1, and y = (— z ,+ 2 ~, y ~ 0

8= x=cscy

MCCCD(Martinez00494

MCCCD/Martinez 01290

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 50 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 71 of 122

MATT 82 T7igonomehy Page 43 of 73 Spring 2010

ThelnverseSecantk~nction y=sec'x 'x5-1 or x>_+1, and B =y=~0, ~t~, y~~

B=x=secy

The Inverse Cotangent T~unction y =cot ' x x ~ (--~,-hw) and 8 = y = (0, ~r )

B=x=coty

Summary of pomain and Range for traverse Trig Functions

y=sue 1x x=[-I,-r1z snd e = y=~-z,+z~

y=cos—F x x=[—1,+l] and B = y=[ 0, ~

y = ~3t]—~ x x = ~—co, -t-oo~ and 8 ~ Y = (— 2' ~ 2 }

y=sec l x, z<_-1 or x>1 and B=y=[0, rt], y~2

y=~s~-'X x_<—i or x>_+L any e =y=~-2,+z~, y~o

y =cot-I x x = (--a~, -l-~) and B = y = (0, n )

MCCGD/Martinez00495

MCCCD/Martinez 01291

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 51 of 243
'•":~'

~:•

~

_

• ~r~

~:

~;.~ a

w~

__ ::.

~'r:

1

Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 72 of 122

MATI82 Trigonomctry Pagc 44 of 73 Springy 201 C

1~X The Inverse Sine, Cosine, and Tangent ~uncfzons Homework

1.. What is the Domain and the range ofy=sinx?

2. A restrtclion on the domain of f ~x~=~x-1~2 to make it one-to-onewould be x=_

3. If the domain of a one to one funcfiaon is ~3,~~, the range of its inverse is

4, True or Fa/se_ The graph of y =cosx is decreasing on the interval (D,s].

~ n

5. fian4= :sin3=

n

6, sinr-- _ ;cosec=

l 6

7. y=sin-~x meaC~s ,where -ISz_<E and -~ <_yS 2.

S. The value of sin ' L

sin 2 ~ is

n

9. cos' - cos- =

5

90. True or False; The domain of y=sin~'x is -Z 5x5 ~ .

11 True or False: sin. sin ' 0~ ~ 0 and eos cos"' Q~ = 0.

12 True or False: y =tan' - x means x = tan y, where -~o < x < ~ and - 2 < y < ~ .

In problems 13 - 23, find the exact value of each expression.

~i 3, ' sin 0 15. sin ' ~-l~ 17. tan"' d 19. sin ' ~ 21. tan' ~ 23. cos ' I z3

l J

For #~ 25 - 35 use a calculator to find the value rounded to two decimal places.

25. siri ' 0.1 27. tan-1 S 29. ' cos g 37. tan-~ ~-0.4~ 33. sin `~-0.12 35. cos ' 3

In problems 37 - 43 find the exact value of each expression. ~Do not use a calculator.

37. sin~sin ' ~0.54~~ 39. cos ` cos 5 I 41. tan~tan^` ~-3.5~~ X43. sin's sin+ - ,~ +

.J l JJ

In problems 45 - 55, do nofi use a calculator. For your answers, also say why or why not, -

45. Does sin'~sinf '~~J -~~

~ 6 6

49. Does cos'~cos~-6 J~=-6?

L 1.1

47. Does sin sin' - (2~' = 2 ?

51. Does coslcos'I -~ f~=- ~

L l 2l 2?

53. Does tan'' ~tanr- 3 ~ + ~? 55. Does tan tan`' ~2~~ = 2?

l JJ

In problems 57 - 63, find the exact solution of each equation.

57. 4 sin-~ tx~ = x 59. 3 cos 1 t2x~ = 2~c 61. 3 tan-1 x = ~t 63. 4 cos ' x - 2~r = 2 cos ' x

MCCCD/N;artinezOD496

MCCCD/Martinez 01292

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 52 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 73 of 122

MATI82 Trigonometry Pale 45 of 73 Spring 2010

1_X2 Values o~~aversc Trig Functions

Find the exact ~talue of sin ' (sin 4 } .

Find the exact value of sin{tan-~ z).

Find. the exact value of cos(sin ! 2)

MCCCDlMartinez0~497

MCCCll/Martinez 01293

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 53 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 74 of 122

MATI82 Trigonamchy Page 46 of 73 Spring 2070

Find fi11e exact value of tan(cos I (— 3))

' Use your calculator to approximate the value in radians fo two decimal places:

a~ sec ~ 3 b) cse^2 (--4}

c} cot-1 ~ z) d} cot' -(-2)

li.

MCCCO/Martinez00498

iVICCCp/M~rtinez 01294

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 54 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10111/13 Page 75 of 122

MATI62 Trigonometry Page 47 of 73 Spring 2010

1.~~ ~aaues of Inverse Trig Functions Homework

1. What is the domain and the ranye of y = secx?

2. Is the graph of y=secx increasing on the intervals ~0,~~ and ~'~,nl?

2 2 ~

3. if cot8=-2 and 0<9<~r, fhen cos9=

4. y = sec`' x means , where ~xI and

. `y `-----~ y ~ 2

5. cos tan' - 1} _

6. 7"rue or false: You cannot obtain exact values for the inverse secant function.

7. Tiue or false: csc"' 0.5 is not defined.

8. True or fa/se: The domain of the inverse cotangent function is the set of real numbers.

In problems 9 - 35, find the value of each expression.

9. cos ~szn t 2 I11. tan cos ' ~- 2 ~

J 13. sec cos`' Z

J

1~ 15. csc~tan~' lj 17• sinCtan~' ~-1~~ 19. sec~sin' - I -1 ~~

l 2

~ 21. ws l~cos ~ ~ 23. suti'~sin(-~ 7 ~J 25. tan~sin'- 3~

~. ~ ~

27. sec tau-1 2~ 29. oot~sin~l f - 3 I~ 31. sin tan 1 ~-3~~

~ ).~

33, sec sin ~ 5l 2 35. sin' cos

J

f n problems 37 - 43, find the exact value of each expression.

l 37, cot' - ~ 39. csc ' ~-1~ 41. sec' 3 2 43. cot"' (- 3 1

Use a calculator to find the value of each expression rounded to two decimal places.

45. sec ~ 4 47. eof~ 2 49. csc`~ ~-3} 51. cot' (~}

.~ 53. csc'I-3~ 55. cot"'I-3~

2 2

_ f (x~ = sinx, and Is (x~ =tan z. Find the exact v lue of each compose function.

57. gl f`'(13 !~ 59. g'~f~ ~ 1~ 61. h(J-'( S~l 63. g

lh,\ 5 ~~ l ~> l J l ~ l

65. g'~f~-3 4 IJ 67. hI g'~-~~

1 l J

.. MCCCD/MartiriezD0499

MCCCD/Martinez OI295

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 55 of 243
Case 2:12-cv-Q0702-DGC Document 68-9 Filed 10/11/13 Page 76 of 122

MAT182 Trigonometry Pa.;e h8 of 73 Spruce 2D10

1.13 Tri~anamefric Identities

Identities and proofs, addition #ormulas, double-angle half-azlgle, redncf~on formulas

Methods used to Sunplify Trigonametrac expressions

Example 1

a) Simplify ~~B by rewrStinE in terzas of sine and cosine functions.

csc 8

cos B 1-sin

b} Show that . _ ~ by mnitiplyzna the numerator &denominator by conjugate

I+sing wsB

of the denominator i.e. 1— sin B,

1+sioB cot9-cos9

c) Simplify + by adding two fractions i.e. getting a common denominator

sin B cos B

d) Simplify 

sin' B-1 by factor➢n~.

tanBsinB-tan8

cot B

a ----

csc B

1

cotB ~ cosB s'tnB_

Solution: a~

~

- - — ---- -- cos 8

csc8 ~ 5 sing I

cosB cose i-sinB cosB(1-siuB) cos~(i-sinB~ b) ~ ~ ~ ~1-sin8

ltsin~ 1+sing 1-sing 1-sin's cos~B cosB

I+sing cots-wsB I+sing cos9 c~tB-cosB sing cosB+sinOcosB+wtBsin9

C~ + ~ ~—+ ~--~

s~ 0 cos B sin 9 cos B cos B sin ~ sin B cos p

cosBt~•rinB cosB+cO56 2co56 2 ~;• ~ Sw 

~ -.~ ~---

sin B oos B sin B cos B sin B ws B sin B

5~' e-1 (see+i?(s~s-i} sli,e+t d~ ~ ~

~" tnn6sinB-tang tanB(sinB-1) tang

MCCCD/~4arti n ezOD500

MCCCD/Martinez 01296

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 56 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/1.1/13 Page 77 of 122

MAT182 Tri~onometiy Paoe 49 of 73 Syring 2010

example 2 Vcrify that csc 0 • tan B =sec 9 yA

Hint: Start with the left side, because it contains the more complicated expression

r Exam ]e 3 Vezify that• sin' ~-B~ +cos' ~-B~ =1 ~ _ P

i

I

i-

~..

Ezample4 Vezify -cosB-sm8

«:

sin ~-8~ - cos ~--9~

~'

~~

n.

TT

~~ ~~l~B

Example 5 Verify that =tan B

l+cot B

a~

ti.-

h9CCCD/MartinezOQ501

MCCCD/Martinez 01297

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 57 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/1.3 Page 78 of 122

MAT182 Trigonometry P~,c SO of 73

sin 6 I+cos B

~+xample6 Vcrifythat 

l+cos9}. sin8 

~ZcscB

taa B +cot B

Example 7 Verify that 

sec B csc D —1

1—sing cosB

Egainple S Verify that =---

casB l+sin6

Spring 2010

MCCCD/MafinezODS02

MCCCD/Martinez 01298

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 58 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 79 of 122

MATl82 Tri~onomelry Pa c 51 of 73 Spring 2010

Guidelines forEstablishingIdentities

1. It is usually best to stazt vrith the more complicated expression.

2. Add oz ~btract fractions.

3. Kewrite one side in tcmis of sine and cosine fimctions only.

4. Always keep in mind the farm of the egression on the other side.

1.13 Tri~onomet~-ie Tdenfities Hame~v4rk

1. True o~ False: sinz B =1-cos"'z 9

2. True or ~afse: sint-B~+cos (-B~ - cos B-sin 9.

` 4, taus B-sec' B.

f_ 5, cos-0~`cosB=

6. True or False_ sin-B~+sin8 =0for any value of 9.

~. Sirrrplify problems 9 - 17 by following the indicated direction.

9. Rewrite in terms of sine and cosine functions: tan~sc9

17. Muitipfy 

°Os9 ~ 1-t-sin8

1-sang 1-+-sinB

13. Rewrife over a common denominator. 

sing+wsB+cosB-sing

cos8 szn~

~sinB+cosB~~sinB+cosO~-I

15. Multiply and simplify: ----

- sin B cos ~

3sin' 8+4sinB+1

17. Factor and simplify:

sing B -I- 2sin 9+ 1

`~ fn problems 19- 703, establish each identify.

19. csc BC~os B = cot B 21. l + tan2 (-B~ =sect B

' 23. cosB{tauB+cot4~=csc~ 25. tan9cotB-cos'B=sin79

27. ~sec8-I~~seeB+l}=tan2 B 29. ~sec8+tanB~~sec6-tanB~=1

31, cos28~l+tanZ B~=1 33, ~sinB+cosB~Z fi~sin9-cas9~Z =2

3S. sec'B-sec~~=tan° B~-tanz 8 37. secB-tangcosh

l+sink

.

39.3si.n'8-h4cas~8=3~-cos~6

Z

,~.-~ l_ cos _ S~p

l+sing

y 43. l+tan6 _ cotB+] 45. secB ~ sin8 _ 2tanB

1-tauB~cotB-1 cscB cosB

47.

I+sing cscB+l

~

-

1-sing cosB

49. ----= =2sec~

1-sin8 csc9-I cosB I -sink

51.

sing _ 1 53 l-sing _~secB-taa4~s

sing-cos4 1-cotB 1+sing

MCCCD/Marttnez00503

MCCCD/Martinez 01299

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 59 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 80 of 122

MA.T182 Trigonometry Page 52 of 73 Sprio~ 20]0

S~ cos9 + sia8 _SIB+casB 57. tangy -H. cosB _secB

1—tan8 l—cot8 l+sin4

X.lQ The Shim a~oEd Difference Foz-muTas

Sum and ]7ifference Forn~nlas for the Cosine Fnziction

'The cosine of the sum of two angles egva}s the cosine of the fast angle times

the cosine of the second angle minus the sine of the 5sst angle times the sine of the second angle.

cos~a-t-~3}=cosacos/3—sinasin~

Memozize these:

cos(a — /J) = cos a cos ,Q +sin a sin Q

~xample 1 Find the exact value of cos75'. Hint 75=45+30

Solution_

4

Example 2 Find the eacact value of cos12 . Hint: lz = a ` s

Solution: ~~~-{-~)

rT l

Vezify tl]e cofunction ~otmula: cost — — ~ (=sin 8 Cofuncbons oFcomplcmevtary angles are equal

lZ f

cos '~ e cos n cos B sin ~ sin B = Q~os B + ll~~in Q = sin B

2 2 2

VeLify the cofunerion fozmula: sin (2 — B} = cos B

Us?n~ the previous cosine cofunctioxi formula i.e, ~cos~ 2 —el = szn 8~ in reverse fives us

J MCCCD/Marfinez00504

MCCCD/Martinez 01300

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 60 of 243
:~~3

;;r n

+Y:

Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 81 of 122

MAT182 Tri;onometry Pale 53 of 73 Spring 2010

sin(2 —B~ = cos ~ ~~ 2 — ~1

~ = cos B thus esYabtishing the cofunction formula.

Sneu and Difference ~+'ornu~las for the Sine ~nncfion

sin ~a +~i~ = sin a cos ~+ cas a sin ~

sin ~a — Q~ = sin a cos ~— cos a sin. /3

Using the eofunetion formula for cosine in reverse gives to

sin~a+/f} = cos(2— ~a+~3)) = cos((2—a)—~)}

= cos(2 —a)cosQ+sin(?--a)sin/~

sin a cos /j + cos cr sin ~

sin (a + ~ ~ = sin acos p -i- cos a sin Q

Using the sum formula for sine gzves us

sin(a — /3} = sin ~ a + (—~)

= sin acos(—~ -1- cos a sin(—/3)

= sin a cos ~ + cos a(--sin f3)

sin(a—~) = sinacos~f—cosasin~

Exaznple 3 Find the exact value ofsin 12

example 4 Find the exact value of sin 80' cos 20' —cos 80' sin 20'

Solurion = ~ (V G +~)

MCCGD/Martinez005D5

MCCCll/Martinez 01301

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 61 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 82 of 122

MATl82 'Cri~onametry Pic ~4 of 73 Spring 2010

Solution =Sin 60' = Y3

2

Esample5 Ifsina=5, a=~~,n~ andsin/j=—~~ /j—(~r, 21

l J

find the exact value of (a) cos a (b) cos fj

~C~ COS (Q -f- ~ ) ~d} SII]~~ ~1- Y

solution: (a) cosy=-3 (b) cos/3=—~ (c) cos~a+~~=~1~ (d) sin~a+Jj~=~2

5 5 25 25

COS ~Q — ~)

~xample 6 Verify that — — cotacot~3+1

s[n a sin ~3

MCCCD/Martinez00506

NICCCD/M:~rtinez 01302

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 62 of 243
Case 2:x.2-cv-00702-DGC Document 68-9 Filed 10/11113 Page 83 of 122

MCGCD/MartinezD0507

MCCCD/M~rtinez 07303

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 63 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 84 of 122

MAT182 Tri3anometry Pale 56 of 73

Proo£ofthe sum. and diLfarene~ formulas far the tangent fuzrctian

tan(a + ~) = 

tan a + tan ~i'

1—tanatan/j

Proof

tan(a+/3}~ sin(a+/j) ^ sinacos~+cosasinQ

cos(a+(3) cosacosJ3-sinasin(3

divide the numerator and tie denominator by cos a cos p

sin~cos/3+c;osasinp

cos a cos ~

ta_n(a + /3} _ — 

cos ~ cos/3 -sinasiu /3

cos a cos ~f

sinacos/3 + cosasin fj

_ cosacos/1 cosacosJ3

sina + sin~3

tan(a+~i) ~ 

Asa cos/j

cos /j sinasin~

cos~3 cosacosQ

tau(a+/j)= tana +tan~3 q.E.D.

1-tanatanJ3

tan a -tan ~3

tan(a-~- l+tanafanQ

~ _.

Proof

tan a +tan(-/~)

tan a - tan(/j)

1 ~- tan a (-tan /~)

cos a cos (3 _ sin asin /j

cosacos(j cosacos/j

Springy 20]0

MCCCD/Martinez005D8

MCCCD/Martinez 01304

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 64 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11./13 Page 85 of 122

_ 'Y

4'! c

MATI82 Trigonometry Palo 57 of 73 Spring 2010

tan(a-/~ = 

tang -tan/j Q~ ~

l+tanafan/j

1.I4 The Sum a~ad ~?iff_erenec Formulas Home~vQrk

1. The distance d from the point (2,-3~ to the point (5,1~ is _

2. if sing = 5 and B is in quadrant II, then cos9 =

3. (a) sin~os ~ _ (b} tan 4 -sin 6 =

4. cos~a+/3~-cos~cos/3~ sinasinQ

5. siu~a-~~=sinacos/j cosasin/j

6. True or False: sin~a+Q~-sina+sin~3-t-2sinasin~3

7. True orFatse: tan 75' =tan30'+tan 45'

8. Irue or False: cos~~-BJ

=cosB

2

(n problems 9 - 19, find fhe exact value of each expression.

9. sin T S 11. cos '7r 13. cos 16S' 15. tanl5' 17. sin 17~ 19. sec - n1

12 12 12 ~ 1?_ 1

In prablems 21-- 29, frnd the exact value ofeach expression.

21. sin2d'cos10'+cos20'sin10' 23, cos70'cos20'-sin7d'sin20'

tan 20' +tan 25' ~c 7z ~r 7n n Sn Srr ~r

25. 27. sin—cos--cos—sin--- 29. cos—cos---+sin—sin1-tan 20' tan. 25' 12 I2 12 12 l2 12 12 12

Find the'exact value of problems 3i-35 under fhe given conditions.

(a) siu(cz+/j) (b) cos(a+Q) (~) sin(a-/j) td) tan (a-/3)

3i. sina-3, 0<a<~i; cosh=~ Z -~<~3<D

5 2 5 2

33. tang=-~, ~<a<~r; cos/3=1 0<~<~

3 2 2' 2

5 3n n

35. sina=l3' 2 

<a<-n; tang=~, 2 <Q<n

37. If sing = ~ , B is in quadrant 2, find the exact value of:

3

td~~l8} 4/

45, sin ~ ~2 + B 1=cos 9

1

47. s;n~~r-4~=sinB 49. sin~~r+B~=-sing 51. tan~~r•-8~=-fang

53, sin~3~-+8 I=-cosB 55, sin~a+~~+sin~a-~~=2sinacos/j

2

In problems 77 & 73, find the exact value of each expression.

(a}cosh (b)sinl8+6~ (c)cosl8-3 j

l l l

In problems 45 -55, establish each identity

MCCGD/Martinez00509

MCCCD/Martinez 01305

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 65 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 86 of 122

TvfA'~182 Trigonometry Page 58 of T3

71. sin~sin ' ~+cos~~ ~~ 73, sin~sin ' ~ —cos ~ ~— 5 f~

1~1

I.15 Daub~e-ankle and Povc~er Reducing Formulas

Double-angle Fortnu(as Powrec Reducing Formulas

sin(2B} = 2 sin B cos B sinZ B ~ 

1— cos(28)

2

cos(2B) = cosZ B—sin' B cos' B 

=1+cos(2B)

2

cos(28) =1— 2 sin 2 B 

~z 8 ;1— cos(ZB)

I+cos(2B)

cos(2~) = 2 cost B —1

tan(2B)- 

2tan0

1— tan1 ~

Qeri~re the formula fw sin(ZB)

Derive the formulas far cos(2f~

Springy 2410

MCCCD/Martinez00510

MCCCD/Martinez 01306

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 66 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 87 of 122

MAT182 Trigonometry Page 59 of 73 S~mng 2010

Derive the fnmiula for tan(2B)

ff sinB=S, B=~Z,n~, find

a) siu(2B) b) cos(2B)

Derive the fn.rmula for sin.z 8

y.,

_. MCCCD/Martinez00511

MCCCll/Martinez 01307

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 67 of 243
Case 2:12-cv-40702-DGC Document 68-9 Filed 10/11/13 Page 88 of 122

MATT &2 Trigonometry Pie 60 of T Spring 2Q10

gerive khe farmufa for costB

Derive the formula for f.2nzB

12ewrite cos° B without any pagers of srte or cosine greater than 1.

MCCCD/Martinez00512

NICCCll/Martinez 01308

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 68 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 89 of 122

MAT182 Trigonometry Pagc 61 of 73

1.15 llo~xble-aa~le anr~ Power~reducin~ I+prmu}as IIQ~txeYvo~-~C

Springy 2010

1. cos(2B~=cosZ B— _ —1=1—

2. sine B = .

2 2

D 1—cosB

3. tan--=

2

4. C3oes cos~2B~have these equivalent forms: cost B—sing B, I-2sin2 B, 2cosZ 8-1?

5. Does sin~2B~ have these equivalent forms: 2sin9cosB and sin'9—cos= D?

k'. ,

6. Does tan~29~+tan~2B~-tan~4B~

!n problems 7 - 17, 0 S B <2n . Use the information given to find the exact value of.•

(a) sin~2B~ (b) cos~2A~ (c) sin (d) cos

-.7. sira8=3,0<B<~ 9. tang=4,n<B<'~ 3 11. cosB=-~,'~ <B<n

~'~~''~=~' ~ 13. secB=3,sinB>0 15. cot8--2,sec6<0 17. tanB=-3,sinB<0

.-~i:' `A'<<'';:~=~-Use Power Reducing Formulas to End the exact value of #19-27.

-,~~ `~19. sin22.5' 21. tan--- 23. cos 16S ZJr. SEC g 27. sin ~ 8

'~.: $

~:~,~j~~'1-4?'~3^•~ iii ~.~ {; `i.: •. ~,

;~ ~,-~:"'+_~~~•~Showthat sin'0-8-2cos~2~~+gcos(4B~

Wit;`{`,.:= i".$c:-'-'. •:3:::

_-~`;~ :- jn' p~blems 47 - 67, establish each identify.

-`;`-?'"~<. ^,.•-_~..'~..-•~' `47•.c CA54~B— S1II~ D = CAS ~2~~

cotZ 8 —1

49. cot~28~= 51. sec(2B)=

sect B 

__~.:a-.~`;i~'-',:i : ,... Z

cos~2B~ cotB-1 B 2

=y;~ _'~`;53.:ctSs? j2~~—sin. ~2B~=cos~48~

x

55. — 57. sec --

-:.:''r ~_;~. ~':~`.;".:., 1+sin~2B~ cots+l 2 1+cosB

-~ - ''- =~~~`::,z;Q: ''sec8+1 Z sin~36~ cos~36~

- ;~~~~'{~~,:59`lcot -- 61.cos~= 63. - -- =2

- ___ <ti`.`~r=-<;:,.:'2 `.sec8-1 e {fib,,: sink cos

l+tangy

~'r'~ ~" ~

~'='"'~'`'."~=' _'3tanB--tan36

:., ~-'65M;~32n 3B

1

67. In sinBl=—(lnll—cos~29~I-1n2)

-~~''' y~~ ~lr ~problerris 69 = 79, frnd the exact value of each expression.

i. <_,~;`~`69_:;sin ism:- ~ 71. cos~2sin' - —~ 73. fan 2cos-I ~--~

'I T``' -'-?~~ - MCCCD/Martinez00513

MCCCD/Martinez 01309

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 69 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 90 of 122

MAT182 Trigonometry Pale 6Z of 73 Spring 2010

75. sin+2cos-'SJ 77. sin'! ~cos'~~ 79. sect 2tan `~~

~ l ~

~.I6 ~ot~rin~Tri~onometric Equations

Example 1: Is 8 = ~ a solution to the equafion sing = ? ? If not, what is the solution?

~xamp(e 2. Solve the equation cosB-2

So1~6on: 8 = 3 +2k~r, B = ~ 5 +2k~r where k is an integer

Example 3: Salve the equation 2sinB-~-f =0, 054<2~r.

MCCCD/M~afinez00514

MCCCD/Martinez 01310

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 70 of 243
:.,:,

Z~f

.;

S'

Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 91 of 122

ZvI,4.7182 Trigonometry Page 63 of 73 Spring 2010

I~nt So(vc for sme. So7urion: x = 3" ~ j or 240, 300

~xampie 4: Solve fhe equation sin 2B= ~ where B= [0,2n).

Soiu!ion: B =z 1 -1- kit, 8 = ii +kit, k = 0, 1, 2 ~ D = {l2' 12 ' 112 ' 112 }

Example 1 Solvethequadratictrinornialequatioa: 2sinZ B-3sinB+1=0, OSB <2~r

MCCCDJMartinez00515

MCCCD/Martinez 01311

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 71 of 243
Case 2:12-cv-00702-DGC DocumenC 6$-9 Filed 10/11/13 Page 92 of 122

` [~

MATI82 Trigonometry Page 64 of 73 Spring 2D10

(~cSnn

~3'mt factor. The solufion set is j 

l6' 6 ' 2

Example 2 Solve the equafion using identities: 3 cos B+3 = 2sin2 B, 0 S B < 2n

I~"w~ Usc Pyd,a~orean Iderrtiiy sinz @ _ ... The solution set is ~~ , n, ~3 ~.

~cample 3 . Solve the equation: cos ~20~+ 3 = 5 cos B, 0 4 8 < 2~c

n Sn

F~znt Reduce 2B . Use double azi;Je formula cos 2B = 2 cos' B —1 The solution set is —

3~3

~zaAople 4 Solve the equation: eos~ B+sing = 2, 0 <_ D <2n

MCCCp/Martinez00516

MCCCD/Martinez 01312

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 72 of 243
s"

Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 93 of 122

MAT182 Tzigonometry Pale 65 of 73

Hint Use Pyt~zgorean Ideality to write in trsms of Sin 9 _ Solution_ sin'" 8 -sin B + l = Q

]..X6 Soivin~ Tri~onomefric Equations

In # 7 - 29, solve each equation on the interval Os D <2~c

Spring 2010

7. 2sin9f3=2 9. 4cos~B=1 11. 2sin~B-1=0

13. sin~3B}=--1 15. cos(29~=-~ 17. sec ~ =~2

19. 2sinB+1=0 21_ tang+1=0 23. 4secB+6=-2

n

25, 3~cosB+2~-1 27_ cos 28--

(

=-1

8 n

29, tan -+- =1

2~~ 2 3

Solve #31-39. Give a general formula for all the solutions. List six solutions.

31, sinB~ 1 33. tang=--~ 35. cosh=0 37. cos(29~=-1 39. sin =-~

2 3 2 2 2

Solve #~41 - 51 for D = [0,2~c). Round answers fo two decimal places.

~41. sinB=0.4 43. tang=5 45. cos9=-4.9 47. secB=-4

49. Stan8+9=0 51. 3sinB-2=0

1. Find fhe real solutions of 4xZ --x-5 = 0.

2. Find the real sofu6ons of x2 -x-1=0.

3. Find the real solutions of E2z-l~~--3~2x-1~ -4=0.

4. Use a graphing utiliiyto solve Sz'-2=x-xz. Round answers to two decimal places.

in problems 5-45, solve each equation on the interval Q_<B <2n.

5.2cos'B+cesB=d 7. 2sin2B-sing-Z=0 9.•~fauB-1~~secB-1~=0

11. sin~B-cos~B=l+cosB 13. sin2 9=6~cos~+1~ 15. cos~2B~+6sin2B=4

17_ cosB=siuB 79. tang=2sinB 21. sing=cscB

23. cos~2B~=cosB 25. sin~7~~-h sin~48}=Q 27. cos~4B~-cos(68~=0

?_9.1+sin8=2cos20 31.2sin'B-SsinB+3=0 .33.31-cosB~=sinZ B

35. tan2B=3secB 37. 3-sing=cost2B~ 41. sia8-~cosB=1

2

43. tan~2B~+2sinB=D 45. sing+cosB=~

In # 47 - 51, find the real zeros of each trig funcfion on the interval 0 _< x < 2n .

47. f~x~=4cos'x-1 49. f~x~-sin~2x}-sinx 51. sinx~-cosx=x

Solve #53 - 63 with a calculator, Round solutions) to fwo decimal paces.

53. x+~cosx=0 55. 22x-17sinx=3 57. sinx+cosx=x

MCCCD/Martinez00517

MCCCD/Martinez 01313

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 73 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 94 of 122

MAT182 Trigonometry Page 66 0€.73

59. xz =z+3cos~2x~ 61. x'-2szn~2x}=3x

1 17 Applications In~oit~in~ Right Triaiz~]es

s~~ 2oio

63. 6sinx—ex =2,x>0

Example 1. If b=2 and a = 40', find a, c, and /j in the ri f t triangle.

~ ~0

7

Solution: a=2tan40'~1,68 and c= 2 X7..61 ,/j=50'.

cos 40'

sample 2: Tf If a = 3 and b = 2, find c, a, and ~3.

c a

3

MCCCD/Mattinez00518

MCCCD/Martinez 01314

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 74 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 95 of 122

-.; >:. .

~~..~.

MATl82 Trigonocpepy Pale 67 of 73

3

Solution: c = 13 ~ 3.b1, a =tan"~ 2 , /3 = 33.7'

Spring 2Q10

Example 3: A sEraight trail leads from the Alpine Hotel, elevation 8000 feet, to a scezdc

overlook, elevation 11,100 feet The length of the trail' is 14,100 feet What is the inclination

(grade) of the trail? That is, what is the angle /j in Figure 47

~ioo fr

_, 3100

solution_ ~3 =sin ~ 12.7° The inclination (grade) of fhe trail is appro7cimately X2.7°

14,100 .

1.X? AppIications In~otvimg Right Triangles Hamewark

1. In a right triangle, if the length of the hypotenuse is 5 and the length of one of the

other sides is 3, what is the length of the third side?

2. True or false; The angles 57.' and 48' are complementary.

3. If B is an acute, solve the equation tan 8 = 2 _ Round answer fo one decimal place.

4. If 8 is an acute angle, solve the equation sing= Z.

5. True or False: In a right triang le, one of the angles is 90' and the sum of the other

two angles is 90'.

6. In navigation or surveying, the from a point O to a point P equals the

acute angle 8 between ray OP and the vertical line through O ,the north —south line.

7. True ar False: In a right triangle, if two sides are known, we can solve the triangle.

8. True or False: In a right triangle, if we know the two acute angles, we can solve the

triangle.

In problems 9 — 21, using the given information, solve the right triangle.

9. b = 5, /3=20'; find a, c, and a 11, a = 6, x=40'; find b, c, and tt

13, b = 4, a=10'; find a, c, and J3 15, a = 5, a=25'; find b, c, and /3

17. c = 9, j~=20; find a, c, and a 19. a = 5, b = 3; find c, a, and /3

21. a = ?_, c - 5; fiord b, a, and /3

23. The hypotenuse of a right triangle is 5 inches. If one leg is 2 inches, find the

degree measure of each angle.

MCCCDlMarfinez00519

MCCCD/Martinez 01315

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.~F

Tvlp1']g2 Trigonometry Pie 68 of 73 Spriug2010

( 24. The hypotenuse of a right triangle is 3 feet. ft one teg is.1 foot, find the degree

measure of each angle.

25. ~ At 10 am on April 26, 20Qb, a building 30~ feet high casts a shadow 50 feet long,

What is the angle of elevation of the Sun?

X18 Laa o~Sixies

Lf none of the an~Ies of a Triangle is a right angle, khe frianQte is called ohliqueAn oblique Mangle will have either three acute angles

oz two acute angles and one obtuse angle (an angle between90' and 180').

Obt~c xi~o

(a} All angles azc acute (6) Two acute ~glcs and one obtvso angle

Tn the discussion that follows, we will alwags label an oblique triangle

so that tt is opposzfe angle a , side b is opposite angle Q , and side c is

c ~ 4 opposite angley.

r

b

To solve an oblique trSangle means to fmd the lengths of its sides and the measurements of

ifs angles. T'o do this, we shall need to lmow cbe length of one side* along with (i} two angles;

(ii) one angle ~d one other side; {iii) the other two sides. `There arc four possibilities to

conszder:

CASE 1: One szde and two angles are Iazowu (ASA oz SAA,).

CASE l: Two sides and the angle opposite one of them are laiown (SSA}.

CASE 3: Two sides and the included angle are Imown (SAS).

CASE 4: Three are [caown (SSS).

The four cases axe aiiusttated below,

~.

a ~ ~ ~

s S 5

Cesc 1: ASA Czse t: SAA Case 2: SSA Case 3; SAS Case 4 : S55

Use the Law of Szxies to solve triangles for Use the Law of Cosines to solve

Case 1 or Case 2. triangles foz Case 3 or Case 4.

Z,a~v of Sines

For a tszangle with sides a , b, c and opposite angles cz,/j, y, respectively,

MCCCp/Martinez00520

MCCCD/Martinez 01316

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Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 97 of 122

NL4T1B2 Trigonometry

c ~ 4

~. r

b

Page 69 of 73

sin a sin ,l3 sut y

a b c

Example X: Solve the triangle: a =40',/3 = 60', a ~ 4

4sin60' 4sin80' 6.I3 Solution: y = g~~, b 'sin 40° ~ 5.~9, C — sin40'

E~mple Z: Solve the triangle: a — 35', J.~ =15', c = 5

I _ -

a

3 s' r

b

Sgrin~ 2010

MCCCD/Martinez005?1

MCCCD/M~irtinez 01317

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 77 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 98 of 122

~~-

MATl82 Tri~onornehy Pic 70 of 73 SP~~2~70

5sizt35" Ssin1S'

solution: y =130', ~ =

sin 130" ~ 

~'~4, b = S~ I~~, ~ 1.69

Example 3: Solve the triangle: a = 3, b ~ 2, a -- 40'

3

' ~~ ~ ~

c

3 sin114.E

solution: sin ~3 ~ 0.4~, y =154.6', c = ~= 4 24

sin40'

.~

,' '~

~ ~ ~~ 6

~6

35 ~~

Eaampl e 4: Solve the triangle: a = 6, b — 8, a = 35'

MCCCD/Martinez0U522

MCCCD/M~rtinez 01318

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 78 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/7.3 Page 99 of 122

F~~:ryr,...;.~.,;_

^ ~ :. ~ ~ MA'C182 Trigonometry Pane 7I o£ 73 Spring 201D

~,~ . ~ solution: p 49.9' or 130.1', 95.1" or 14.9' c ~ 10.42 or c ~ 2.69 ,~ Qz~ Yt ~ Y~^' ~ i~ z

~.

_ 1 Eaample S: Solve the tziangie: a = 2, c ~ 1, y = 50'

a=~

~. .

_ Q,

~:.

b

`::-. ',

'~" ~~ ~~ Solution: sina =2sin50' .=:1.53

:~' ''.118 X.aw of ~iaies Hon~.e~or~

~. 1. The difference formula for the sine function is sin~a—~~=

-~ - s; = 2, if B is a acufe angle, solve the equation cosB= 2

'~ .3. A triangle with sides 2 and 5 is similar to a triangle with corresponding sides of

rf,; 3 and x .Find the missing lengths, .

4. If none of the angles of a triangle is a right angle, the triangle is called

5. f=or a triangle with sides a,b,c and opposite angles a,~3,y,

the Law of Sines states that

6. True of False: An oblique triangle in which two sides and an angle are given always

results in at feast one triangle.

7. True of False: The sum of the angles of any triangle equals 180'

!n problems 17 — 23, solve each triangle.

17. a-40', /3=20',a=2 19. ~=70', r=10', b=5

21. rz=110', r~30', c=3 23. a=40', (3=4Q', c-2

MCCCD/Marfinez00523

MCCCD/M~rtinez 01319

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 79 of 243
Case 2;12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 100 of 122

. "~MAT182 Trigonnmeuy Pie 72 of 73 Spring 2010

:' ' • In problems 25 — 35, iwo sides and an angle are.given. Determine wF~ether the given

inforrnafion results in one Triangle., tvra triangles, or no triangle at all. Solve any

~,~ V ~ fnangle(s) that results.

` 25. a=3,~b =1,a=50' 27, b=5, c=3, ~=100" 29. a=4, c=6, ,6=20`

~~ 31. b=4, c=6, ~i=20' 33. a=2, c=1, y=100' 35. a~2, c=1, y=25'

1.19 haw of Cosine

We used the Law of Sines to solve Case 1 (SAA or ASAj and Case 2 (SSA) of an oblique

trian;te. NoW we use the Law of Cosines to solve Case 3 and Case 4.

Case 3. Two sides and the includ'zng angle are known (SAS).

Case 4: Threw sides are known (SSS},

Law of Cosines Theorem

For a triangle with sides q b, c and opposite angles a, ,D, y respectively,

cZ =a~~b~-2abcosy ~ (1)

I1~=Q2-f-C2-24'C CAS ~z~

~xauzpie 1 Solve a SAS Triangle where a = Z, b = 3, y = 60".

f~

c

60- o.

3

MCCCD/Martinez00524

MCCCD/Martinez 01320

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 80 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 101 of 122

IvIt1T182 Trigonometry Pagc 73 of 7i Spring 2010

solution_ c = ~1 ~ , a ~ 40.9' , /3 ~ 79.1'

~~"~.'' ' example 2 Sotve a SSS Triangle where a = 4, b = 3, c = 6.

~ 6

~.. Y

3

Solution: a .~: 363', Q ~ 26.4', y =117.3'

1.19 Law of Cosznes ~ozne'c~o.rk

1. Write the foRnula for the distance dfrom P =(x„ yl ~ to PZ =(xz, y~~,

2. (f B is an acute angle, solve the equation cosB= ~ .

3. If three sides of a triangle are given, the Law of is used to solve the

triangle.

4. ff one side and two angles of a triangle are given, the Law of is used fn

solve the triang{e,

5. If two sides and the included angle of a triangle are given, the Law of,___^__

is used to solve the triangle.

6. True or False: Given only the three sides of a triangle, there is insufficient information

to solve the triangle.

7. True or False: Given two sides and fhe included angle, the first thing to do to solve

the triang{e is to use the Law of Sines.

8. True or False: A special case of the Law of Conies is the Pythagorean Theorem.

In problems ? 7 -- 31, solve each friangle.

MCCCDRvtartinez00525

MCCCD/Martinez 01321

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 81 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 102 of 122

MAT182 Tri~onomctry Pic 74 of 74 Springy 20]0

~.. 17. a=3 b=4, y=~10" 19.b=1, c=3, a=BQ' 21. a=3, c=2, a=80'

23.•a-2; b=2, Y=5D' 25. a=J2, b=I3, c=5 27. a=2, b=2, c=2

29. a=5, 6=8, c=9 31. a=XO, b=8, c=5

~~:

.:.-.' .

MCCCD/Martinez00526

MCCCD/Martinez 01322

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 82 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page x.03 of 122

MCCCD/Martinez 01323

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 83 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 104 of 122

Exhibit B

MCCCD/Martinez 01324

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 84 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 105 of 122

[F';.d: (Fwd: Re: Printing of Course Ma'tcrial]] .

Subject: [Fwd: [Fwd:.F2e' Printing of Course Material]]

From: f'aut DeRose <paul.derose@pcmail.maricopa.edu>

Qate: Thu, 02 Sep 2010,5:29:11 -0700

To: Casandra ltakar <casandra.kakar@pcmaii.maricopa.edu>

1 of 1

Subject: Re: Printing of Course Material

from: Ronnie Elfiatt <ronnie.eiliott@pcmail.maricopa.edu>

Data: Tue, 12 Jan 2010 14:14:21 -07Q0

To: "cleopatria.martinez" <cleopatria.marfinez@pcmail.maricopa.edu>

MAT 182 and 187 are the two I'm aware of at this point.

c]eopatzia.martinez wrote:

Please remind me, what aze the items Z had prinked fog fall that were suspect

and the sane foz'my spring requests,

Ronnie Elliott wrote:

i~Cleopatria,

As you may or may not know, we have mechanisms in place to red flag potential

copy right issues. Last week it was brought my attention that some of the

items you had printed for fall were suspect and the same for your spring

requests. Do you have any document of approval fron the publisher to use their

material? ZP so, would you please provide that documentation to me ,o that S

can release your requests to be printed. If you do not have such approval,

them Ne will not be able ~o print youz requests at this time.

Ronnie

iRe: Printing of Course Material.eml 

Content-Type: messagelrfc822

Content-Encod i ng: 7bit

2 EXHIBIT NO.

o ~~SY6~V

MCCCD/Mattinez01130

MCCCll/M~►•tinez 01325

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 85 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 10/11/13 Page 106 of 122

Exhibit C

MCCCD/Martinez 01326

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 86 of 243
Case 2:12-cv-007p2-DGC Document 68-9 Filed 10/11/13 Page 107 of 122

r ofz

o ~autts~r rto. ,~~Q

Subject: Printing ofCovrse Maluial `

Fram: Ronnie Elliou<ronoicctliottepcmail.maricopa.cdu> ~~~

Date. Tuc, 26 Jan 2010 07:52:16 -0700

To: Clcopulria Mar[inci <deopairia.manineza(~ypcmail.maricopa.cdu>

CC: Ronnie Elliot( <ronnic.dliori~Jpcmaii.maricopa.edu>, Lcc Combs dec.coNbs@damail.maricopa.cdu>, Margaru McConnell 

<mazgarctmcconncilQdomail.mericopa.cdu>, Anna Sollcy Sanna.sollcy npcmail.maricopa.edu>, Casandra Kakar

~casandr6.kakarulyanaiLmaricopa.cdu>

Clcopalria,

[n ordu to respond to your cmait dated Janvary 20, 2010, 1 svlicitcd the guidance oCour district Jcgal counstl, Margaret McConnell,

regarding copyright infringement and fair use as defined in Ccdcral law. !did prwid~ Macgarei with coyies of your requests and z ropy of

the adopted teat book, and this morning « ccivcd I~cr findings bsscd on the federal copyright law, including tht limitztions of F it use.

The Uistrict'5 Adtrtinistrative Regulation 3.2 provides in pari as follows:

4. E»~loyees arc prohibited front copying inate~~ials noI spccificalh' aflomed b~.16c (IJ copyr-iohl7mv, (7J fair ~ece guideli~)cr. (J)

licenser orcottlracr~al agree~nutu. or (4J other prrn~istiw~.

S. The Governing Board disapprn~rs of unmuAorccd duplicaunn in a~~~~jm~n. L•mplo~~eet x•6o Sri/Ifii!!7~ dirreoard this Board policy

and/or the ajm~enrentioned cop~~righ~ guidelines do so n~ !heir o~i~rt risk and ❑ssu,ne ull liahilrh~ jot ihci~ acrioxs.

The focus here is whelhor your use of eenain materials mea.s ont of the criteria specified in Para 4 gnorcd above, We dmi'( have any

information that suggufs that your ust its wi~hin the specific exception under the ~opyngh~ !nw authorizing the use of copyrighted

materials without permission Nor have we seen any licenses or wriltcu ycmiission from tlr copyright holder, Pwrson, oFlhe materials

for use of dieit inattria;s in your lecture roles. Therefore, we must focus on "fair use."

The doctrine of fair use allows for the use of copyrighted works without khe

owner's permission. It protects limited uses of copyrighted works from being an

infringement. The doctrine is, however, determined on a case-by-case basis.

The Federal Copyright Act provides that the "fair use of a copyrighted work for

purposes such as criticism, comment, news reporting, teachiny (including

multiple copies for classroom use), scholarship, or research, is not an

infringement.of copyright," Again, not ail educational uses are fair use. Four

factors are considered in determining whether the use of a work is a fair use:

i. the purpose and character of the use, including whether such use is of a

commercial nature or is for nonprofit educational purposes;

z: the nature of the copyrighted work;

3. the amount and substantiality of the portion used in relation to the

copyrighted work as a whole; and

a. the effect of the use upon the potential market for or value of the copyrighted

work.

17 U.S.C. ~ 107. Courts decide fair use by weighing each factor individually and

then determining which way the combinakion of factors tips.

In light of those legal standards, we

have reviewed your January 20 o-tnail and d~c following r.~atcrials - 3 documcut cnuttcd "Phoenix College Falb 20D9, A4AT 16?

Trigonometry Section, Lccturc Notes, Dc Clcopelria Manincz" and a book, Precxlculut: Contorts Thro~h functions: A R~~ht

'Triangle Approach to Trieonometrv, Inswctors Edition, authored by Michael Sullivan and Michael Sullivan, IlJ. The book indicaus

that the copyright holder is Pearson £duca4on, Inc and ii saysr "All rights reserved_ No part of this book may be reproducccl, in any Form

nr by any morns, without permission in writin, from the puSlisher.'

The first issue here is whethu your lecwre noes are original wori:s or problems created by you, or whether porlions are instead copied

from the Prccalculua book or semc other copyrishtcd souroc

In reviewing parts of your notes and the book, we have some rez! Concerns. Hnached arc some pages of your no~cs on which W c hzvc

EEOCOO~~~o t o a: i z pNr

MCCCD/Martinez 01327

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 87 of 243
Case 2:12-cv-00702-DGC Document 68-9 Filed 1Q/11/13 Page 108 of 122

handwriftrat die page oumbcrs of the Prec:. us

book as the source of the cxampla, tcxi or problem. Once we camc up wish that 10 or so correlarions, we sioppcd looking, as [he

problerc seemed char. The examples or problems are copied vubatim in;o [hc lecture notes.

Addjtionally, ~hcrc is no attribution in each of these instance of die source of d~c w:amplt or problem, that is, no identification of Pearson

as the copyriglri hofdcr_ WhUhcr you intend to or not, the Icdurc notts give the distinct impressign tha[they arc your original wo: k,

which is inco~recl. That lack of alviLucion is, by i~sclf, a 1cgaPissue. As to what the breadth of the problun is, only you know that

because we don't know all of Utt resources that you may have used for thost notct.

Moreover, ii is our understaodirtg that you may Dave been using thrse notes over a period of lime,

As you can see from the listing of the "fair use" rcquircmcnts, Uic use must not Uc one tbat, if it became widuprcad ovu an entire market,

would reduce the poccntial market For the copyrighted ~~rork. to this case, copying problem or example from the book, with the potential

that the students don't need to purchase it, would make your use widiou! permission a problem.

Od~cr guideline devdopcd ender tfic auspices of~he fedcal govcmmca[ specify that use must'sponpneou5." It is hard to m3ke.a case

that this is spontaneous use iCyou use the materials over and over each scmesmr. In any event, it would appear to have taken a significant

effort to copy the problems and ezampla into the teat of your lecture notes, ilicreby eliminating any "spontaneity."

Based on die above, we bdicve that you need to get wrictcn permiscion to use portions of the book in your lecture notes from Pearson,

and make sure that you note that Pearson is tlx copyright holder of materials. Publishers such as Pearson hove spocific stiff dedicated to

this issue. You should find outwl~o those arc, idcnfi(y the portions of the book [hat you wish to use, and obtain pumission in writing.

Bolrom line, Cleopacria, you must secure writlrn permission from she publisher before we arc able ro print your material.

Regards

Ronaio

Lecture Notes and Book.pd( 

~onlenl-Typt: ~ application/pdC

Content-Enrnding: base64

ot~ z EEOCOt~h~'i~~ n a: ~ ?Pry

MCCCD/Martinez 01328

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 88 of 243
__ 

~ ~ t •

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 89 of 243
Martinez vs MCCCD

Videotaped Deposition of Cleopatria Martinez, Ph.D. - 7/10/2013

UNITED STATES DISTRICT COURT

DISTRICT OF ARIZONA

Cleopatria Martinez,

Plainti:Ef,

u CV12-00702-PHX-DGC

Maricopa County Community College

District,

Defendant.

VIDEOTAPED DEPOSITION OF CLEOPATRIA MARTINEZ, Ph.D.

(VOLUME 1, Pages 1 through 187)

Phoenix, Arizona

July 10, 2013

9:08 a.m.

Prepared by:

Janet Hauck, RPR

Arizona Certified

Reporter Number 50522

GLENNIE REPORTING SERVICES, LLC

602.266.6535 - www.glennie-reporting.com

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Martinez vs MCCCD

Videotaped Deposition of Cleopatria Martinez, Ph.D. - 7/10/2013

77

A. Mm-hmm.

Q. -- would you clip the documents together?

Would you put them in a file folder? How would you store

these documents?

MR. STROJNIK: Object to the form of the

question.

Go ahead.

THE WITNESS: In whatever form they were

given to me, I would keep them.

Q. BY MS. BALCH: Do you recall ever having

applicant information in manila envelope -- in manila

folders?

A. Yes.

Q. Do you recall wYiether manila folders had the

applicant names on them?

A. I don't recall.

Q. Please take a look at paragraph 15 of your

complaint. States, "Defendants have accused, criticized,

and reprimanded Professor Martinez regarding the content

of her lecture notes." What are these lecture notes that

you're referring to? Can you please describe them for

me?

A. They are the guide that I used for teaching.

Q. What do they typically look like?

A. They describe whatever the topic is that I'm

GI~ENNIE REPORTING SERVICES, LLC

602.266.6535 - www.glennie-reporting.com

MCCCD/Martinez 01203

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Martinez vs MCCCD

Videotaped Deposition of Cleopatria Martinez, Ph. D. - 7/10/2013

78

teaching, and they develop the concept and provide

examples.

Q. Are they typically bound, stapled, clipped?

A. Do they come -- I present them both ways.

Q. Do they contain a cover page?

A. Some do.

Q. They contain table of contents?

A. Yes, some do.

Q. Page numbers?

A. Yes.

Q. How many pagEs are these lecture notes,

typically?

A. They vary. I could have just a couple of

lecture notes, or I can have up to -- pages or up to 100

or 150 pages. Some -- they vary.

Q. Please explain all -- strike that.

Can you please explain all facts that you

have in support of your contention that MCCCD has

criticized and rep.r_imanded you regarding the content of

your lecture notes as .referenced in paragraph 15 of your

complaint?

MR. STROJNIK: For..m.

THE WITNESS: Paragraph 15?

Q. BY MS. BALCH: Correct.

A. I -- I've already given you some. Actually

GLENNIE REPORTING SF'RVICFS, LLC

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Martinez vs MCCCD

Videotaped Deposition of Cleopatria Martinez, Ph.D. - 7/10/2013

106

Q. -- in a week?

A. Yes.

Q. Did you meet al_1 of these students?

A. Yes.

Q. What did you have these students do?

A. They made copies for me. They typed material

up for me. They did research. They stapled. They

collated. They transcribed what I had written. They ran

errands when I needed something taken somewhere to

another colleague or a department.

Q. Anything else?

A. Not that I can think of.

Q. Did they do any filing for you, any data entry

for you?

A. No.

Q. You mentioned that they typed materials for

you. What materials did they type?

A. My notes. I'd write notes, I -- my lecture

notes, or tests when I'd write them.

Q. What do you mean they would type notes? How

would they type the notes?

A. On a computer, they would type.

Q. But would you -- what would you give the

students so that they could type up your notes?

A. A written document that they would type up for

GLENNIE REPORTING SERVICES, LLC

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Martinez vs MCCCD

Videotaped Deposition of Cleopatria Martinez, Ph.D. - 7/10/2013

107

me.

Q. Describe that written document. Was it a

piece of notebook paper --

A. Yes --

Q. -- that you had written on?

A. -- that I had written on.

Q. And what were these materials that they were

typing?

A. They could be tests, quizzes, handouts, my own

lecture notes, lists of formulas, any handout that I

wanted them to have -- the students to have. That's --

that's it. My syllabi, sometimes T'd have them type up

my syllabi.

Q. Did you ever provide your work study students

with a textbook and instruct them to --

A. Yes.

Q. Okay.

A. To -- I'd say in my notes I'd refer to the

problem in the textbook that I wanted inserted at a

point.

Q. Do you remember what textbooks you utilized in

this manner?

A. Just a Sullivan & Sullivan.

Q. And the materials that these work study

students were workinq on, what courses were those for?

GLENNIE REPORTING SERVICES, LLC

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Videotaped Deposition of Cleopatria Martinez, Ph.D. - 7/10/2013

108

MR. STROJNIK: Objection as to form.

Are we still in fall 2009, or are we

discussing all work study students that she's always had?

I remember you began with fa11 2009.

4• BY MS. BALCH: Mm-hmm. Yeah, we were in fall

2009, so in fall 2009, what --

A. Oh, okay, okay. So what was the question?

Q. 1n fall of 2009 what -- strike that.

In fall of 2009 were you having your work

study students type lecture noes and class materials for

you?

A. Yes.

Q. What classes were these lecture notes and

course materials for?

A. Whatever I was teaching fall of 2009, tests,

quizzes, and handouts.

Q. Did they type answer keys for you?

A. No, I don't think so.

Q. Did these work study students perform any

additional tasks?

A. They made -- I've already named all the tasks.

Q. Other than what you've told me so far, did

these work study students perform any other tasks for

you?

A. Not that I recall.

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MR. STROJNIK: From --

THE WITNESS: Go ahead.

MR. STROJNIK: From the spring of `09 --

THE WITNESS: Okay.

MR. STROJNIK: -- through the present,

other than this document, which is Exhibit 3, marked MAT

082, Spring 2010 -- other than this document, did you use

any other -- in any other courses, did you solely use

your lecture notes as the textbook?

THE WITNESS: No, I did not. Just this

one.

Q. BY MS. BALCH: Prior to January of 2009, had

you ever used your lecture notes solely as the textbook

for a math course?

A. No.

Q. flow did you go about creating these lecture

notes?

MR. STROJNIK: Object to the form of the

question.

Q. BY MS. ~ALCH: Do you understand my question?

A. Which lecture notes are you asking about?

Q. ~'o.r example, let's take MAT 082 --

A. Okay.

Q. -- the one labeled, Chapters for Basic

Arithmetic. It's a voluminous document --

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A . Mm-hmm .

Q. -- right?

A. Mm-hmm.

Q. It`s --

A. Yes.

Q. -- comprised of -- looks like 105 pages. Does

that look about right to you?

A. Yes.

Q. I'm assuming this took quite a bit of time to

prepare?

A. Yes.

Q. How did you prepare this document?

A. Well, over the - years I've written this

information many times, and so some of it was already

written that I had written in previous years, my previous

40 years, and some I sat down and I wrote them. I

organized them, and X -- I had material from many, many

different people that T used to put this together.

Q. Did you type this document?

A. Yes, I did.

Q. The entire document, or did you have

assistance in typing this document?

A. I believe I typed this one. I may have had my

work study students -- I may have given them pages that

were already written, and ttzey typed it. They typed

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those pages, but I don't remember which pages those are.

Q. So let me make sure I understand this

correctly. Your testimony is that either you typed this

document --

A. Mm-hmm.

Q. -- or in instances where you might have had

work study students type portions of this for you, you

would have, what, written it out on a scratch piece of

paper and handed i~ to the work study students for them

to type?

A. Yes, that's right.

Q. Did you provide any textbooks to your work

study students to copy problems from to insert into the

lecture notes at Exhibit 3, titled, Chapter for Basic

Arithmetic?

question.

MR. STROJNIK: Object to the form of the

You can answer if you can.

THE WITNESS: No textbooks were used.

Q. BY MS. BALCH: No textbooks were used?

A. I credited the people from whom I had

documents on the thank you page.

Q. But on the credit page there are no publishers

of textbooks, correct?

A. Correct.

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Q. And is it your contention that you or work

study students acting on your behalf did not copy any

math problems from a textbook to be inserted into these

lecture materials?

MR. STROJNIK: Object to the form of the

question.

THE WITNESS: State it again, please.

MR. STROJNIK: Can you add "to your

knowledge" to that question?

MS. BALCH: Certainly.

Q. BY MS. BALCH: Actually, I'll ask -- T'l1_ ask

it in two separate pars.

A. Okay.

Q. That should make things a little bit easier.

When you typed -- back up. So this

document was created in one of two manners. Either you

typed the document and/or you assigned to your work study

students certain pages that you had handwritten and told

them to type up, correct?

A. Yes.

Q. Sa for the portions that you typed

personally --

A. Mm-hmm.

Q. -- did you copy, or borrow, or verbatim

utilize any math problems that you found in math

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textbooks?

A. No, absolutely not.

Q. As far as the portions that you gave to your

federal work study students to type, in your handwritten

notes that you provided to the work study students, did

you indicate ~o your federal work study students that you

wanted them to copy certain problems from certain

textbooks to incorporate into the lecture notes they were

typing?

question.

MR. STROJNIK: Object to the Form of the

You can answer if you're able to.

THE WITNESS: No.

Q. BY MS. BALCH: You understand my question? I

know it's getting late in the day, but do you understand

my question?

A. Say it again, please.

Q. Earlier today you testified that when work

study students would type your lecture notes for you,

sometimes you would put a note in your handco.r_itten sheets

where you would ask the students to insert math problems

from a certain textbook; do you recall us discussing that

earlier today?

A. Yes.

Q. When you were preparing the lecture notes .f..or

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Math 082 that are at Exhibit 3, when you were handwriting

out the pages that you wanted your federal work study

students to type, did you instruct them to copy math

problems from textbooks?

A. No, absolutely not.

Q. Do you still have those pages --

A. No.

Q. -- of handwritten notes?

A. Oh, no, that was four or Live years ago. No.

Q. When was the last time that you created

lecture notes of the type that are found in Exhibit

Number 3?

MR. STROJNIK: Object to the form of the

question.

THE WITNESS: I'm not sure if you're

asking me for a bundle of them, or one page of them,

or -- what do you consider lecture notes2

Q. BY MS. BALCH: T -- I'm using your term,

lecture notes. And when I've asked you to describe what

your_ lecture notes are, I think we can agree that you'd

characterize these three packets at ExYlibit Number 3 to

be lecture notes, correct?

MR. STROJNIK: I believe she testified

these were textbooks and lecture notes.

THE WITNESS: Right.

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the students do not have to buy the textbook?

MR. STROJNIK: Object to the form of the

question.

Go ahead.

THE WITNESS: No, T -- I just teach -- I

teach the content of the subject, and I select what I

think is best in terms of content and what's .relative and

understandable, best understood by my students, what is

the best product for my class. That's how I select a

textbook.

Q. BY MS. BALCH: Do you always assign a textbook

in your courses?

A. Yes. Yes, I do.

MR. STROJNIK: Form.

THE WITNESS: I do.

MS. BALCH: Just want to make sure we have

a clear record of that.

THE REPORTER: Can we go off the record

for a second?

1:29 p.m.

THE VIDEOGRAPHER: Off the record at

(Brief recess.)

THE VIDEOGRAPHER: On tYie record at

1:25 p.m.

Q. BY MS. BALCH: Dr. Martinez, I just want to

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make sure that we have a clear record here. There was a

little bit of confusion on the record, and I think that

some of us spoke over each other, so I'm going to ask

this question again.

Da you always assign a textbook in all of

the courses -- in all of the math courses that you teach

at Phoenix Community College?

MR. STROJNIK: Object to the form of the

question.

THE WITNESS: T assign what I call a

textbook in all my courses.

Q. BY MS. BALCH: And what do you call a

textbook?

A. A series of pages that has the content that

I'm teaching.

Q. Are some of these textbooks your .lecture

notes?

A. They have been.

Q. For what classes?

A. Are we -- in which time frame?

Q. Let's talk from January of 2009 to the

present,

A. January -- and what are you asking me again

about that time frame?

Q. Classes where the assigned -- where the

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assigned textbook is your lecture notes.

A. In January of 2010?

Q. January of 2009 through the present.

MR. STROJNIK: She's asking starting

January 2009 for which of your classes did you use your

lecture notes as the textbook?

THE WITNESS: Oh, okay. It was for a -- I

believe it was a college algebra class.

Q. BY MS. BALCH: Do you recall the course number

for that college algebra r_ou.r_se?

A. No, I don't.

Q. Any other courses?

A. From January of 2009 forward -- January?

Q. Yes.

A. I may have used my notes f_or an arithmetic

class, Math 108.

Q. 108?

A. Mm-hmm -- oh, no wait, I'm sorry, not 108.

Math 082, 082.

Q. Any other courses?

MR. STROJNIK: Do you have any -- I'm

sorry, do you have any documents that could possibly

refresh her recollection? Because I know there's a ton

of documents that we can look at that would help her.

MS. BALCH: I -- I -- I have some

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documents, but for courses that she's -- well --

MR. STROJNIK: Or if you have them by

memory, maybe you can ask them specifically.

MS. BALCH: Sorry, Counsel, I don't have

that in my fingertips.

MR. STROJNIK: Okay. Do you have that in

your memory bank, because you could just ask he.r -- I

mean, it's at center stage pretty much, but go ahead, to

the best of your recollection.

THE WITNESS: I don't -- you know -- T

don't remember what I was teaching. That was four or

five years ago. I don't remember, but I believe I taught

the two classes T mentioned using my lecture notes.

MR. STROJNIK: She's asking from January

2009, fall, spring -- fall, 2009, spring, fall, 2010 and

on and on --

THE WITN~S~: Right, right.

MR. STROJNIK: -- if you can --

THE WITNESS: Well, I -- I was precluded

from using notes after spring 2010. I haven't been able

to use notes since then.

(Exhibit No. 3 was marked.)

Q. BX MS. BALCH: You've been handed what's been

marked as Exhibit 3. Exhibit 3 consists of three packets

of documentation.

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A . Mm- hmm .

Q. One -- one is labeled Chapters for Basic

Arithmetic?

A. Mm-hmm.

MR. STROJNTK: Is that a yes?

THE WITNESS: Yes.

Q. BY MS. BELCH: The next is marked at the top,

MA`T 182 Trigonometry, Spring of 2010; do you see that?

A. Yes.

Q. And the final is labeled, MAT 182 Precalculus,

Trigonometry Section Lecture Notes for fall of 2009; is

that correct?

A. Yes.

Q. Do you recognize these documents?

A. Yes.

Q. What are these documents?

A. They are my lecture notes for the subjects.

Q. And do these packets of lecture notes appear

to be true and accurate representations of the lecture

notes that you utilized in these courses?

MR. SI'ROJNIK: T'm going to -- in light of

the allegations in this lawsuit, I would recommend that

you thoroughly review these exhibits to ensure they are

accurate.

MS. BALCH: Take your time.

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MR. STROJNIK: Yes, take your time. This

is important.

If you'd like, so we don't use up your

time, do you want 'to do this off the record so we don't

use your seven hours? Because I am going to recommend

that -- there's copyright allegations in this lawsuit.

We have to make sure that each page is actually the

actual page and something isn't inserted in there

unbeknownst to us. I'm not saying intentionally, but

maybe mistakenly by someone.

MS. BALCH: Absolutely, Counsel. How

about this? Can we agree to stipulate or to -- to

basically stipulate to whether or not these are true or

accurate representations after you've had the opportunity

to review them?

MR. STROJNTK: Sure, okay.

MS. BI~LCH: And that's Exhibit 3, all

three of these packets, so

MR. STROJNIK: So we're reserving the

r9_ght later to say that these are not accurate.

THE WITNESS: Okay.

Q. BY MS. BALCH: And with that caveat, these

generally appear to be the types of lecture notes that we

have been discussing; is that correct?

A. Yes.

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Q. And these are the types -- and at least for

two of the courses that you have taught since January of

2009, these are the types of materials that you used as a

textbook?

A. Yes.

Q. And one of those courses was Math 082?

A. Yes.

Q. And this first document in this packet, it's

Bates numbered MCCCD/Martinez 00528 through 636. If you

flip to the third page of that packet, do you see at the

top it says, MA2 -- MAT 082, Spring 2010 outline?

A. Yes.

Q. While -- subject, of course, to you and your

counsel's review of this document on a page-by-page basis

to ensure that there are no inadvertent documents that

have been omitted or inserted, does this packet appear to

be the tExt that you t~t.i_lized for Math 082?

A. Yes.

MR. STROJNIK: Object to form.

THE WITNESS: Yes.

Q. BY MS. BALCH: end was this the only assigned

textbook that you utilized for your Math 082 course in

spring of 2010?

A. I di_d use this in 2010? See, I'm not sure

what I taught in 2010.

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Q. Welt, I'm -- I'm asking you, Dr. Martinez, and

at the top of the third page --

A. Mm-hmm.

Q. -- of that packet --

A. Oh, okay.

Q. -- it says --

A. Oh, I see.

Q. O.r wait, one, two, three. It says MAT 082,

Spring 2010 Outline?

A. Yes.

Q. So does this document appear to be the

textbook that you utilized in spring of 2010 for your MAT

082 course?

A. Yes.

Q. And other than this packet, did you utilize

any other assigned textbooks for your MAT 082 spring 2010

course?

A. No.

Q. Let's take the next packet. At the top it

says -- I'm looking at the one that says, MAT 182

Trigonometry.

A. Mm-hmm, yes.

Q. And the Bates range is MCCCD/Martinez 00400

through 00450.

A. Yes.

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MR. STROJNIK: You know what Bates is,

right? Okay.

Q. BY MS. BALCH: Does this appear to be the

lecture notes that you utilized in your spring of 2010

MAT 182 Trigonometry course?

A. Yes.

Q. And was this the only assigned textbook in

that course?

A. No.

Q. What other textbook was assigned?

A. I don't remember, but there was another

textbook.

Q. How many assigned textbooks?

A. One.

Q. Do you recal]_ either the name or the author of

the textbook?

A. No, I don't.

MR. STROJNIK: Oh, boy, looking at this

stuff, I'm glad I'm through with all of this.

Q. BY MS. BALCH: Last packet is labeled, MAT 182

Pr_eralculus, Fall 2009.

A. Mm-hmm, yeah, yes.

Q. And the T3ates number at the bottom is

MCCCD/Mar_tirlez 00451 through 526.

A. Yes.

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Q. Do -- does this packet appear to be the

textbook that you utilized in your MAT 182 course in fall

of 2009?

A. Yes.

Q. Did you require any other assigned textbooks

for that course?

A. T do not recall.

Q. And earlier you testified that you believed

you used lecture notes similar to those contained in

Exhibit 3 as the textbook Lor at least two courses -- two

math courses from January 2009 to the present. One was

Math 082. The other one was College Algebra. Is that

correct?

A. Okay, now that I've got these documents --

Q. Mm-hmm.

A. -- I wasn't sure about that. Would you go

through the question again?

Q. Absolutely. From January 2009 through the

present, what courses did you teach where you used your

lecture notes as the textbook for the course?

A. As the textbook?

MR. STROJNIK: Object to the form of the

question.

Q. ~Y MS. BALCH: Let me clarify. What -- for

the classes that you taught from January 2009 to the

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present, please identify for me the courses that you

taught where the -- where your lecture notes, similar to

those in Exhibit 3, were the only textbook --

~. Oh, okay.

Q. -- that you utilized for the course?

A. That would be the Math 082, spring 2010.

Q. ire there any other courses --

A. No.

Q. -- where -- let me -- let me finish the --

A. Oh, okay.

Q. -- the question first.

Are there any other courses from January

2009 thr..ough the present where the -- your lecture notes

were the only assigned textbook for the course other than

MAT U82?

~. And the timeline was?

Q. January of 2009 through the present.

A. Where my lecture notes, is that what you said?

Q. Your lecture notes, similar to those in

exhibit 3 to your deposition.

A. Okay. This i.s the only one, the Math 082.

MR. STROJNIK: Do you understand the

question?

THE WITNESS: I'm -- I think I do. Shalt

I say i.t, because I'm not sure. I'm not sure.

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Vi..deotaped Deposition of Cleopatria Martinez, Ph.D. - 7/10/2013

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MR. STROJNIK: From --

THE WITNESS: Go ahead.

MR. STROJNIK: From the spring of '09 --

THE WITNESS: Okay.

MR. STROJNIK: -- through the present,

other than this document, which is Exhibit 3, marked MAT

082, Spring 2010 -- other than this document, did you use

any other -- in any other courses, did you solely use

your lecture notes as the textbook?

THE WITNESS: No, I did nod. Just this

one.

Q. BY MS. BALCH: Prior to January of 2009, had

you ever used your lecture notes solely as the textbook

for a math course?

A. No.

Q. I-Iow did you go about creating these lecture

notes?

MR. STROJNIK: Object to the form of the

question.

Q. BY MS. BALCH: Do you understand my questi.on?

A. Which lecture notes are you asking about?

Q. ~'or example, let's take MAT 082 --

A. Okay.

Q. — the one labeled, Chapters For Basic

Arithmetic. It's a voluminous document --

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A . Mm- hmm .

Q. -- right?

A. Mm-hmm.

Q. It's --

A. Yes.

Q. -- comprised of -- looks like 105 pages. Does

that look about right to you?

A. Yes.

Q. I'm assuming this took quite a bit of time Lo

prepare?

A. Yes.

Q. How did you prepare this document?

A. Well, over the years I've written this

information many times, and so some of it was already

written that I had written in previous years, my previous

40 years, and some I sat. down and I wrote them. I

organ.ized them, and I -- I had material from many, many

different people that I used to put this togethe.r_.

Q. Did you type this document?

11. Yes, I did.

Q. The entire document, or did you have

assistance in typing this document?

71. I believe I typed this one. I may have had my

work study students -- I may have given them pages that

were already written, and they typed it. They typed

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Videotaped Deposition of Cleopatria Martinez, Ph.D. - 7/10/2013

134

those pages, but I don't remember which pages those are.

Q. So let me make sure I understand this

correctly. Your testimony is that either you typed this

document --

A. Mm-hmm.

Q. -- or in instances where you might have had

work study students type portions of this for you, you

would have, what, written it out on a scratch piece of

paper and handed it to the work study students for them

to type?

A. Yes, that's right.

Q. Did you provide any textbooks to your work

study students to copy problems from to insert into the

lecture notes at Exhibit 3, titled, Chapter for Basic

Arithmetic?

question.

MR. STROJNIK: Object to the form o~ the

You can answer if you can.

THE WITNESS: No textbooks were used.

Q. BY MS. BALCH: No textbooks were used?

A. I credited the people from whom I had

documents on the thank you page.

Q. But on the credit page there are no publishers

of textbooks, correct?

A. Correct.

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135

Q. And is it your contention that you or work

study students acting on your behalf did not copy any

math problems from a textbook to be inserted into these

lecture materials?

MR. STROJNIK: Object to the form of the

question.

THE WITNESS: State it again, please.

MR. STROJNIK: Can you add "to your

knowledge" to that question?

MS. BALCH: Certainly.

Q. BY MS. BALCH: actually, I'll ask -- I'll ask

it in two separate parts.

A. Okay.

Q. That should make things a little bit easier.

When you typed -- back up. So this

document was created in one of two mariners. Either you

typed the document and/or you assigned to your work study

students certain pages that you had handwritten and told

them to type up, correct?

A. Yes.

Q. So for the portions that you typed

personally --

A. Mm-hmm.

Q. --- did you copy, or borrow, or verbatim

utilize any math problems that you found in math

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Martinez vs MCCCD

Videotaped Deposition of Cleopatria Martinez, Ph.D. - 7/10/2013

136

textbooks?

A. No, absolutely not.

Q. As far as the portions that you gave to your

federal work study students to type, in your handwritten

notes that you provided to the work study students, did

you indicate to your federal work study students that you

wanted them to copy certain problems from certain

textbooks to incorporate into the lecture notes they were

typing?

MR. STROJNIK: Object to the form of the

question.

You care answer i.f_ you're able to.

THE WITNESS: No.

Q. BY MS. BALCH: You understand my question? I

know it's getting late in the day, but do you understand

my question?

A. Say it again, please.

Q. Earlier today you testified that when work

study students would type your lecture notes for you,

sometimes you would put a note i_ri your handwritten sheets

where you would ask the students to insert math problems

from a certain textbook; do you recall us discussing that

earlier today?

A. Yes.

Q. When you were preparing the lecture notes for

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Videotaped Deposition of Cl~opatria Martinez, Ph.D. - 7/10/2013

137

Math 082 that are at Exhibit 3, when you were handwriting

out the pages that you wanted your federal work study

students to type, did you instruct them to copy math

problems from textbooks?

A. No, absolutely not.

Q. Do you still have those pages --

A. No.

Q. -- of handwritten notes`?

A, Oh, no, that was four or five years ago. No.

Q. When was the last time L- hat you czeated

lecture notes of the type that are found in Exhibit

Number 3?

MR. STROJNIK: Object to the form of the

question.

THE WITNESS: I'm not sure if you're

asking me for a bundle o.f_ them, or one page of them,

or -- what do you consider lecture notes?

Q. BY MS. BALCH: I -- I'm using your term,

lecture notes. And when I've asked you to describe what

your lecture notes are, I think we can agree that you'd

characterize these three packets at Exhibit Number 3 to

be lecture notes, correct?

MR. STROJNIK: I believe she testified

these were textbooks and lecture notes.

THE WITNESS: Right.

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Videotaped Deposition of Cleopatria Martinez, Ph.D. - 7/10/2013

138

Q. BY MS. BALCH: Okay. What -- what would you

like to refer to the documents at Exhibit Number 3 as?

I'll use whatever term you --

A. This is the -- the textbook.

Q. This is the textbook, okay. Have you

created -- since spring of 2010 --

A. Mm-hmm.

Q. -- have you created any new textbooks of the

type that are -- that are at Exhibit 3?

A. No.

Q. Since spring of ?_009, what courses have you

created textbooks for?

A. This is the only one, the 082.

Q. So in courses where you create these types of

materials

A. Mm-hmm.

Q. -- but you also have a separate assigned

textbook by a publisher, do you still refer to these

notes as your textbook?

MR. STROJNIK: Object to the form of the

question.

THE WITNESS: These notes meaning the 082

notes or --

Q. BY MS. BALCH: In general, so these notes --

A. Okay.

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Martinez vs MCCCD

Videotaped Deposition of Cleopatria Martinez, Ph.D. - 7/10/2013

139

3.

Q. -- at the -- the other two packets at Exhibit

A. I do not call them textbooks. They are

lecture notes.

Q. So the distinction is, if I understand this

correctly, if -- with respect to the documents that you

prepare to be utilized as course materials --

A. Mm-hmm, yes .

Q. -- if it is the only course materials, you

refer to it as a textbook?

A. Yes.

Q. If it is designed to be used in addition to a

textbook from a publisher, you refer to them as lecture

notes?

A. Yes.

Q. Why did you create your own textbook for MAT

082?

MR. S`I'ROJNIK: Object to the form of the

question.

THE WI'T'NESS: Because -- I -- I created

it, because I knew what the content was that students

needed to learn, and I -- I know how I present it. I

think I pr..esent it better in a more clear fashion. This

is more customized to the students at Phoenix College

in -- in vocabulary, and presentation, and in order,

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Martinez vs MCCCD

Videotaped Deposition of Cleopatria Martinez, Ph.D. - 7/10/2013

140

and -- and in breadth, in depth, whereas textbooks are --

don't -- I didn't think fit the students as well. So I

created my own textbook.

Q. BY MS. BALCH: Did you create the textbooks in

order to help students save money?

A. I -- I -- I just answered why I created them,

because of the content and the presentation, the quality

of the questions, the presentation, the order, the

development of the concept. I have many years of

experience that has shown me some techniques and problems

that help students -- that motivate students and help

them learn best, and I wanted a textbook that reflected

the best.

Q. So is it your testimony that trying to help

students save money did not have any bearing --

MR. STROJNIK: Asked and answered. Come

on.

Q. BY MS. BALCH: -- on your decision to create

these lecture notes?

MR. STROJNIK: Object to the form of the

question.

Don't answer that. You've answered this

three, four times already.

Move on to the next line of questioning.

Q. BY MS. B11LCH: Dr. Martinez, let's talk about

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Martinez vs MCCCD

Videotaped Deposition of Cleopatria Martinez, Ph. D. - 7/26/2013

UNITED STATES DISTRICT COURT

DISTRICT OF ARIZONA

Cleopatria Martinez,

Plaintiff,

v. CV12-00702-PHX-DGC

Maricopa County Community

College District,

Defendant.

VIDEOTAPED DEPOSITION OF CLEOPATRIA MARTINEZ, Ph.D.

VOLUME 2 (Pages 188 through 278)

Phoenix, Arizona

July 26, 2013

11:12 a.m.

Prepared by:

Janet Hauck, RPR

Arizona Certified

Reporter Number 50522

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Martinez vs MCCCD

Videotaped Deposition of Cleopatria Martinez, Ph.D. - 7/26/2013

223

can see the date that was on there, but my best -- T

would think -- I think it was after these.

MS. BALCH: Thank you.

A11 right. Do you want to take that break

real quick, Counsel --

MR. STROJNIK: Sure.

MS. BALCH: -- confer with your client?

THE VTDEOGRAPHER: Off the record at 12:02.

(Recess from 12:02 p,m. to 12:08 p.m.)

THE VIDEOGRAPHER: On the record at 12:09.

Q. BY MS. BALCH: Dr. Martinez, do you know who --

I might mispronounce his name, but Johnny Santellan is?

A. Yes, I do.

Q. Who is he?

A. He's an adjunct math instructor.

Q. Would you consider him to be a friend?

A. Yes, I would, a friend and a colleague.

Q. Do you know what .r_ace he is?

A. No.

Q. Do you know what national origin he is?

A. I believe he's Mexican.

Q. Have you ever asked him to make photocopies

for you?

A. Yes.

Q. When?

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Martinez vs MCCCD

Videotaped Deposition of Cleopatria Martinez, Ph.D. - 7/26/2013

224

A. I don't recall what the date was.

Q. What did you ask him to make photocopies of?

A. Over a weekend I had written up a page of

inverse trig function graphs because the book didn't have

them all together, and it didn't show them all, and I

wanted my students to know what they looked like and have

information about them. So I spent the entire weekend

writing this one page up and putting the information the

way I thought -- the best way I thought it would be, and

Monday morning I -- but I did it over the weekend, and

Monday morning I wanted a copy, but I couldn't find

Joe Sueyoshi, and I couldn't find Casandra Kakar, and I

wanted to use it in my class.

And I knew it wasn't a copyright

violation, because I had written it myself that weekend,

so I thought since it's not a copyright violation because

I wrote it, I -- I know that, because I just did it alt

weekend long it took me. I asked Johnny if he would make

a copy, acid I would -- I was going to hold them until Joe

said okay, because it was so close to classroom time, and

if he didn't say okay, then I wasn't going to use them.

Q. Did you ever get the photocopies back from

Mr. -- is it Santellan?

A. Santellan.

Q. Santellan. Did you ever get the photocopies

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Martinez vs MCCCD

Videotaped Deposition of Cleopatria Martinez, Ph.D. - 7/26/2013

232

Q. Did you ever ask him to use his course

materials for one or more of your classes that you were

going to be teaching in fall of 2012?

A. Several times.

Q. What did he say?

A. In 2012?

Q. Yes.

A. Ask the question again, please.

Q. Did you ever ask him to use his course

materials for any classes that you were teaching in the

fa~1 of 2012?

A. Yes.

Q. What classes and what did he say?

A. Right now, I don't remember. The class was --

he gave me permission. He said he -- spring -- no, I

don't .remember what I taught spring or what I taught fa1.1

this year, so I can't remember. I'm getting the two

mixed up.

Q. Did you use those materials for the courses

that you were teaching?

A. Yes, Z used it as a textbook.

Q. Did you distribute materials to students?

A. I did.

Q. How di_d you distribute them to students?

A. We_11_, at the beginning of the semester --

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Videotaped Deposition of Cleopatria Martinez, Ph.D. - 7/26/2013

233

because this was a last minute change -- I asked the

students whether they wanted to make their own copies,

and I needed to have them done quickly, because this was

already the first or second day of classes, or whether

they wanted me to make the copies and then they would

reimburse rrie for whatever it cost me to make the copies.

And that way, we would all have them at the same time.

And all but one student said they wanted me to make them,

and then the next class period, the one remai.ni_ng -- one

student said, no, you make mine, too, please.

So I went to Staples, and I had copies

made and bound. I came back and I took them to class,

and I told the students at first it would be 20 -- about

$20. I did not know what it was going to cost. And i.t

turned out to cost a little bit over $11, and T told them

$11. And so I told them that they need to reimburse me

for the -- and they agreed, so we exchanged the book for

the $11.

Q. And did the students pay that money directly

to you?

A. Yes.

Q. Did you submit -- strike that.

MR. STROJNIK: That's a good question

actually -- ask.

Q. BY MS. BALCH: This copy job that you processed

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Videotaped Deposition of Cleopatria Martinez, Ph.D. - 7/26/2013

247

change or not, because the grade is still A.

Q. Dr. Martinez, you used to be the math

department chair; is that correct?

A. Yes.

Q. What were your job duties and responsibilities

as a math department chair?

A. I represented the math department faculty

interests to the administration, and I communicated

messages from the administration to the math -- to the

math department. I scheduled -- T wrote -- wrote the

schedule each semester. I -- I dea]_t with any complaints

from students or among faculty. I supervised the

chair -- the -- the secretary, and all the workers, work

study, the students, et cetera. I made sure the

department ran smoothly and that the needs of the faculty

were heard by the administration.

Q. Did you have the ability to hire anyone?

A. No, the chair does not hire.

Q. Did you have the ability to Fire anyone?

A. No, the chair does not have -- fire.

Q. Did you have the ability to demote anyone?

A. No.

Q. Or reduce anyone's amount of pay?

A. The chair does not have that authority.

Q. Was it -- would you cha.r.acterize it as more of

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Martinez vs MCCCD

Videotaped Deposition of Cleopatria Martinez, Ph.D. - 7/26/2013

248

an administrative position?

MR. STROJNIK: Object to the form of the

question.

Answer if you'ze able.

THE WITNESS: Faculty don't necessarily

consider that administrative, although that might be

changing, so I can't answer it right now.

Q. BY MS. BALCH: Dr. Martinez, are you claiming

that t~iere was some sort of policy or widespread practice

that existed that caused you harm?

MR. STROJNIK: Object to the form of the

question.

Answer if you are able.

THE WITNESS: Would you restate that

question?

Q. BY MS. BALCH: Xeah. Are you clai.minq that

there was a policy or a practice that existed that was

discriminatory?

A. A policy?

MR. STROJNIK: Object to the form of the

question.

Answer if you're able to.

THE WITNESS: I can't address the -- I

don't know about policy.

Q. BY MS. BALCH: Do you know what a policy is?

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Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 129 of 243
C^~

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Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 133 of 243
Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 134 of 243
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deci,~u.aLs, i.v~.teacrs, rv,r~ rati~v~.al ~.uru.bers; Frrnortioi~~, av~.d percev~.tac~es.

othertapi~s ~y~c~,tide represew~-ati~ws aFdata, geovuetric ~i.gur°s, avid HuEaskre~x.%wtMCCCDlMartinez0052B

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 135 of 243
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~l Gi V~.{2 l~ D u.

to the fo~l.ow~u,g colleagues for shar~~.g their ~~.sp~ratiow, their ideas, tHe~r cov~,~.w~.ewts, ~he~r

suggestiows, awd their w~r4~ which prrnluted the eov~tev~.t -~rthis vutttcriaC.

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pr. ~vu.a~~a v~~~egas

MCCCD/Martinez00529

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1V~AT 082 Spring 2010 Outline

Basic Ari~hmetxc

Due Date Chapter 1— Numbers ~ Operations Page

1.1 Types of Numbers &Vocabulary --------------- 1

1.2 Place Value -------------------------------------------- 11

1.3 ZZounding Numbers --------------------------------- 12

1.4 Adding ----------------__--------------------------------- 13

1.5 Subtracting -------------------------------------------- 14

1.6 Multiplication -------____..----------------------------- 16

1.7 Long Division ---__________..-------------------------- 1$

1.8 Divisibility Tests -------------------------------------.. 21

1.9 Checking Division ---------------------------------..-- 23

1.10 Exponents ----------------__-------------------___------- 24

1.11 Order of Operations --------------------------------- 26

1.12 Mean, Median, Mode --------------------------------- 29

Due Date Chapter 2 — F.ractions Page

2.1 Reducing Fractions ---------------------------- 33

2.2 Adding &Subtracting Fractions ----------- 36

2.3 Multiplying &Dividing Fractions ---------- 39

2.4 Mixed Numbers -------------------------------- 41

2.5 Operations wifh Mixed Numbers ---------- 43

2.6 Addition and Subtraction of Fractions ---- 45

2.7 Multiplication &Division of Fractions --- 46

2.8 Mviced Fraction Problems -------------------- 47

2.9 Ratios ---------------------------__------____-------- 48

MCCCD/Martinez00530

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Due Date Chapter 3 —Geometry Page

3.1 Points, Lines, Planes, &Angles ------------- 52

3.2 Two-Dimensional Figures -------------------- 55

3.3 The Pythagorean Theorem-------------------- 59

3.4 Three-Dimensional Figures------------------- 62

3.5 Perimeter ------------------------------------------ b4

3.6 Area -------------------------------------------__--_ 67

3.7 Volume -_-_--------------------------..____--------- 72

Due Date Chapter 4 — Intesrers (i.e. Signed Numbers) Page

No HomeworLc 4.1 Number Line &basic concepts ---------------- 76

4.2 Adding Signed Numbers_______________..______..__ 76

4.3 Multiplying and Dividzng Signed Numbers 79

4.4 Order of Operations ------------------------------- 81

Due Date Chapter 5 — Decimals ~ Percents Page

5.1 Basics about Decimals ----------------------------- 84

5.2 Changing Decimals, Fractions, Percents ----- 88

5.3 Adding &Subtracting Decimals ---------------- 9~.

5.4 Multiplying Decimals --------------------------- 92

5.5 Dividing Decunals --------------------------------- 93

5.6 Percent Problems. ---------------------------------- 94

5.7 Signed Numbers &Critical Thinking --------- 96

Due Date Chapter 6 — Metrie System, Graphs, ~ Page

Unit Conversions and Graphs -------___________ 9S

Sununation ------------------------------------------- 102

HOMEWORK ANSWERS --- 104

Final Exam wilt be May 13, 2010, in Room B210

8:30-9:45am ~'liursday

MCCCD/Martinez00531

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MAT Q82 Basic Arithmetic Page 1 of 105 Spring 2010

Chapter 1—Numbers and ~~erations

1.1 7y,~es ofJVumbers and Vocabulary

I7ay 1: Whole Number, Mixed Number, Negative Number, Positive DecimaC, Negative Decimal,

Improper Fraction, Proper Fraction, Natural Number, Integer

MCCCD/Martinez00532

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MAT 082 Basic Arithmetic Page 2 of 105 Spring 2010

The following vocabulary will be covered in the next eight days as follows:

Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8

A-C D-F I-M N-R R-W Symbols Complex

Even/odd

Jl7atli Y'ocaba~clar,~ N''ords

Absolute Value is the distance from zero of a number inside vertical baxs.

I-7I is 7 because -7 is 7 units away from zero

Addends are numbers being added. addend

+ addend

sum

Day 9

Powers

Roots

The Associative Pro~~ has to do with grouping using parentheses (2+5),

brackets [2~+-5], or braces {2+5}. The associative property says that when adding or

multiplying you can group different numbers without changing the value of the answer.

Example: (2+5)+8 is the same as 2+(5+8)

The average of some numbers is found by adding all the numbers and dividing the

answer by however many numbers were added. Another word for average is the mean.

Example: the average of 2, 5, and 8 is (2+5+8) divided by 3 which is 15/3 or 5

The base in the number 32 is the 3. 3z means 3 times 3 which is 9.

Commutative Pxoperty has to do with changing the ozder of adding or multiplying.

The commutative property says that changing the order of numbers does not change

the value of the answer.

Example: 2+5+8 is the same as 5-+-8+2

Composite numliers are numbers that can be written as the multiplication of 3 ar more

numbers, Example: 6 is (1)(2)(3) so 6 is a composite number Howevex, S is not

composite because 5 is (1)(5). Five is called a prime number.

FYI: the number "1" is the only number which not prime and not composite either.

A Constant is a number and a variable is a letter.

Example: 7 is a constant and y is a variable

Cubed means multiplied times ztself three times.

Example: 5 cubed is written 53 and this means (5)(5)(5) which is 125

MCCCD/Martinez00533

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MAT 082 Basic Arithmetic Page 3 of 105 Spring 2010

Distribute means to multiply more than one time.

Example: 3(4 + 5) means to nnultiply the 3 times the 4 and the 3 times the 5, fihen add

those two numbers.

3(4 + 5) = 3(4) + 3(5) = 12+15 = 27

The denominator is the number written at the bottom of a fraction and this is khe

number That gives the fracfion its name. The top number is the numerator which tells

you how many you have.

Example: In 20 the 5 is the denominator. n~erator ~- dividend

5 denominator E-- divisor

Dimensional analysis is the procedure to change from one unit of measurement to

another by using fzactions whose value is "1" like 

1 foot or 3 feet or 5280 feet

12 inches 1 yard 1 mile

Example: Change 5 £eet into inches.

5 feet = 

5 ft..12 in. _ s . 12 in. = 60 inches

1 1 ft.

The dividend is the number you are dividing into. quotient

T'he divisor is the number you are dividing by. divisor dividend

The quotient is the answer of a division problem.

Evaluate means give the value. Example: Evaluate x + 2 when the x is 7.

Answer: The value of x+ 2 is 9.

An E~monent is the small number telling you how many times to multiply the base.

(The exponent is written above and to the right of the larger number called the base)

3~ f—cxponen( i.e. 3 to the 4th power is (3)(3)(3)(3) which is 81

aos~

A factor is something multiplied times something else.

Examples: 9x5 9.5 (9)(5) axb C•D ~x~~y~ Sx

To factor means "write as multiplication'

Example: Factor 45 means to write it as (5)(9) oz (3)(15) or (3)(3)(5)

MCCCDlMartinez00534

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MAT 082 Basic Arithmetic Page 4 of I OS Spring 2010

An imvra~er fraction has the top number (numerator) larger than the bottom number

or the same size as the bottom number.

Examples: 5 13

5 7

Index index radicand Example: 3 64 3 is the index

16 is 2 16 2 is the index

Integers are all the positive numbers, the negative numbers, and the zero.

Integers are ... -6, -5, -4, -3, -2, -1, 0, +I, +2, +3, +4, +5, .. .

Invert the fraction means to flip numerator and denominator.

Example: If you invert 3 you get ~ which is also called the "reciprocal".

LCD stands for Lowest common denominator.

LCM stands far least common multiple. LCM means the same as LCD.

Example: The LCM for 2, 5, 6 is 30 The LCD for ~ + 5 -~- 6 is 30.

Each number can be multiplied to get 30: 2(15)=30 5(6)=30 6(5)=30

Also, 30 can be divided evenly by each number: 30 30 30

2 S 6

The mean of some numbers is found by adding all the numbers and dividing the

answer by however many numbers were added. Another word for mean is the average.

Example: 5 is the mean of 2, 5, and 8 since (2+5+8) divided by 3 is 15/3 or 5

The median is the middle number in a list of numbers listed in order of size.

Example: 8 is the median of the numbers 2, 2, 3, 6, 7, S 19, 32, 39, 40, 75

The minuend is the number you are subtracting £rom. amend

subtrahend

difference

A mixed number is a whole number with a fraction like 2 B .

You can change a mixed number to a fraction by multiplying 8 times the 2

then add the 3 so you get the fraction 19

8

The mode of some numbers is the number that is repeated the most times.

For example: 28 is the mode for 3, 9, 14, 17, 3, 28, 19, 28, 6, 5, 28

MCCCD/Martinez00535

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MAT 082 Basic Arithmetic Page 5 of ]OS Spring 2010

Natural nnrnbers are the numbers 1, 2, 3, 4, 5, 6, 7, 8, .. .

They are also called "counting numbers."

Negative numbers are the numbers less than zero.

The numerator is the number written at the top of a fraction and this is the number that

tells us how many you have. The bottom number is the denazninatar which tells you

the name of the fraction.

20 numerator F- dividend

Example: In 5 the 2Q is the numerator.

denominator E-- divisor

Positive numbers are the numbers greater than zero.

The Power of a number is the exponent, 3Z is zead "3 to the second po~cver."

The "2" is the power of the "3."

A grime number can be divided by no other number except 1 and itself.

Examples of prime numbers: 2, 3, 5, 7

FYI: the number "1" is the only number which not prime and not composite either.

Prime factorization is factoring by using only prime numbers

A roduct is the answer to multiplication. factor

x factor

factor times factor =product

product

A groper fraction has the top number smaller than the bottom number.

Example of a proper fraction: 6

7

A quotient is the answer to division.

numerator dividend 

quotient

= quotient divisor dividend

denominator divisor

T1ze radicand is tine part written inside the radical.

"°" radicand 3 g "8" is the radicand

MCCCD/Martinez00536

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MAT 082 Basic Arithmetic Page 6 of 105 Spring 2010

Rational numbers are numbers that have repeating decimals.

Examples: 2 is .500 0 8 is .37500 0 3 is .666 6 7.572323 23

These are not rational numbers

(i.e. irrational numbers): ~ Tr .573733733373333

The reciprocal is the inverted or flipped fraction.

Example: The reciprocal of g is 4

The square root is represented by this radical symbol ~ radicand

The part inside is called the radicand. The square root of 49 = 49 = 7.

The square root of a number is the number which times itself is the radicand.

Example: 49 is 7 because 7 times 7 is 49 which is the radicand

Square numbers are the result of multiplying a number times itself, Examples of

squared numbers are: 0, 1, 4, 9, 25, 36, 49, 64

The subtrahend is the number that is being subtracted in a subtraction problem.

minuend

c~i}~trahPnr~

uzjjerence

A variable is a letter and a constant is a number.

Example: 7 is a constant and y is a variable

Whole numbers are the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, .. .

Notice that "0" is a whole number but it is not a "natural number."

✓Ylal`1t Y'ocabular~ in Symbols

< is less than 5 is less than or equal to

> is greater thau >_ is greater than or equal to

= is equal to ~ is appz~oximateIy equal to

a~solnte value ! faetox~ia,~

~— clivision

Lhe square root of

the cube root of

;~ the fourth root of

suinmatioxz (add)

~' pr ~ 3.14 or

e ~ 2.718

i ~ iunaginai•y uuznber

is parallel to

l is perpeuclicular to

MCCCD/Martinez00537

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MAT 082 Basic Arithmetic Page 7 of 105 Spring 2010

COMPLEX Numbers a + bi

Real Numbers

Rational Numbers

Integers Not-Integers

-3,-2,-1,0,1,2,3,... i.e. l/2 .7 3.1

WL-iole Numbers

0,1,2,3,...

Natural Numbers

1,2,3,..

Irrational Numbers

Ima,~inar~ Numbers

Even Nunnbers are numbers ending with 0, 2, 4, 6, or 8. They can be divided evenly by 2.

Odd Numbers are the rest of the numbers: 1, 3, 5, ,

List the first 6 even numbers

List the first 6 odd numbers

List the first 6 prime numbers

List the first 6 composite numbers

MCCCD/Martinez00538

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MAT D82 Basic Arithmetic Page 8 of 105 Spring 2010

Properties of Real Numbers

If two fractions are added together, do you always get a fraction?

If two real numbers are added together, do you always get a real number? Closure Property for Addition

If two fractions are multiplied, do you always get a fraction?

If two real numbers are multiplied, do you always get a real number? c1o5~e rroPerry Eor M~pli~ation

Can you add real numbers in any order? like 3-+-5+9 or 5+3+9 Commutative Property of Addition

Is the answer the same with both additions?

Can you multiply real numbers in any order? like 3(5)9 or 5(9)(3) Commutative Property of Addition

Can you subtract real numbers in any order? like 9 — 5 or S — 9 Axe the answers the same?

Can you divide real numbers in any order? like 8 = 2 or 2 -~ 8 Are the answers the same?

What is 12 divided by 3? How do you check this division?

MCCCD/Martinez00539

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MAT 082 Basic Arithmetic Page 9 of l05 Spring 2010

What is 0 divided by 5

What is 12 divided by 0?

What number can you add to a number and the first number is the answer? Zero--the Additive Identity

~lat C1I1 you TTlultlpl~ t1ITles a IlumbeI' and ~lE f1I'St I~U117beI 1S ~1e anSWe1~ One—the Multiplicative identity

What do you add to a number to get the answer zero? [ts opposite—The Additive ]nverse

What do you multiply times a number to get the answer one? lu inverse—The Multiplicative Inverse

However, the number zero does not have a rrultipiicative inverse.

What happens when you multiply a number times one? Why?

What happens when you add zero to a ntunber? Why?

What happens when you multiply a number times zero? Why?

the Multiplicative ]dentity

the Additive Identity

the Multiplication Property of Zero

What happens when you multiply a number times one? Why? the Multiplicative identity

The last vocabulary words we will talk about are numbers called "squares," "cubes,"

square roots, and cube roots.

MCCCD/Martinez00540

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MA"i' 082 Basic Arithmetic Page ] 0 of 105 Spring 2010

Square

Numbers Cubes Square roots &Cube Roots

02 O 0 03 z 0= 0 3 0 = O

12 1 7 13 2 1 = 1 3 1 = 1.

z~ 4 s 23 ~4 = 2 3 8 = 2

3~ 9 27 33 ~9 = 3 3 27 = 3

42 16 64 43 Z 16 =4 '64 = 4

5 2 25 125 5 3 z 25 = 5 ' 125 = 5

62 36 216 63

7 2 49 343 7 ~

8z 64 512 83

92 81 729 93

1oz goo ~,000 zo3

112 121

122 144

132 169

1.1 7'ria~es of.Numbers a~zd Y'ocabular~ Hoxne~vo~•lc

Identify the number as Whole Number, Mixed Number, Negative Number, Positive Decimal,

Negative Decimal, Irrzproper Fraction, Proper Fraction, Natural Number, Integer

1. 1 S 2. 8 4 3. 0 4. 3.781 5. 83,001

6. -8 7. 61 8. 9 ld. 33.7 11. -5

I2. 457 13. 5 14. l 4 15. -14.1 16. 5.8

17. 4 18. ,o I9. 362,049 20. OI 21. 7 B

Z2. 33.7 23. 24. 25. TC 26. 75,039 27. 3

28. -87 29. 6.49 30. -.5 31. 2,0 32. 'g

33. Which are even numbers? 3, 14, 9, -8, 71, 50, n, e, 4'-, 25 , (-7~ , 3b

MCCCD/Martinez00541

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MAT 08Z Basic Arithmetic Page 1 ] of 105 Spring 2010

1.2 Whole Number Place Value Chart

0 0 :~ b

*' 0 ~ O ~ ~

G

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1 1 1

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Trillions

r

Billions

~

Millions Thousands Ones

Each gxoup of three A comma is written

digits has a name. between each group.

1.2 Place Value Homework

Identify the place value of the digit 2 in each number.

1. 61,284 2. 82,110 3. 284,100 4. &23,415

5. 725,837,166 6. 44,265,199 7. 253,045,701,000 8. 823,000,419,567

Name the place value far each 0 in this numbez: 302,016,450,098,570

9. The 1st zero 10. The 4th zero _

11. The 2nd zero 12. T'he 5th zero _

13. The 3rd zero

Name the place value for each 0 in this number: 81Q,704,069,809,035

14. The 1st zero 15. The 4th zero

16. T'he 2nd zero 17. The 5th zero

18. The 3rd zero

Write each number in words.

19) 8421 2Q) 1936 21) 46,205 22) 75,089 23) 3,064,801 24) 7,900,408

Translate the words into numbers.

25. Forty-six thousand, eight hundred five. 26. Seventy-nine thousand, forty-six

27. Five million, six hundred thousand, eighty-two 28. One million, thirty thousand five

29. Two hundred seventy-one million, nine hundred thousand

30. Three hundred eleven million, four hundred

Which of the numbers {~, 3 , —4 5 , —'3 , a , 10 , ~, 72, E, I, —65, Si} are:

31. Natural numbers

32. Integers

33. Rational numbers

34. Whole Numbers

MCCCD/Martinez00542

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MAT 082 Basic Arithmetic Page 12 of 105 Spring 2010

1.3 Ro~.ndin~ Numbers

Steps for rounding numbers

Step 7 Locate the number that is being rounded. Replace the numbers after that number

with zeros.

Step 2 Look only at the first number you are changing to zero.

If this number is 4 or less, the rounded number stays the same.

If this ntunber is 5 or more, add 1 ~o the rounded number.

Step 3 Use the ~ symbol to indicate that the rounded number is now an approximation

(close, but not exact).

Example 1: Round 14.39652 to the nearest thousandth. (Is it closer to 14.396 or 14.397?)

1.3 Rounding ~umhers Homework

Round to the nearest ten:

1) 45 2) 85 3) 661 4) 123

Round to the nearest hundred:

5) 16,462 b) 12,799

Round to the nearest thousand:

9) 38,431 10) 117,011

Round to the nearest hundred thousand:

13) 5,254,423 14} 7,395,999

~} 823,402

11) 12,577

15) 9,007,601

8) 701,529

12) 677,529

16} 3,116,201

MCCCD/Martinez00543

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MAT 082 Basic Arithmetic Page 13 of 105 Spring 20I0

1.4 Basic O~erafiions: Adding

There are 4 basic operations: Addition, Subtraction, Multiplication, and Division.

negative numbers on this side positive numbers on this side

O

This line is called the number line.

It has all the negative numbers on the left of zero and all the positive numbers on the right.

Numbers which are larger than zero are called positive numbers.

Positive numbers can be written with a "+" sign in front of them ox without any sign at all.

Saving 4 dollars and saving 11 dollars is like adding two positive numbers. This would be a

+4+11 4+11

savings of 15 dollars. In numbers this looks like this: 

+1S ~r 15

Negative numbers are smaller than zero and they must be written with a minus sign in front.

If you spend 8 dollars at one store, then spend 6 dollars at another store, you spent 14 dollars.

In numbers this looks like this: —6 — g

—14

Add these positive numbers:

13+8+2+21+6

Add these negative numbers:

—13-8-2--21-6

1.4 Adding Homework

Perform the following addition.

1. 41 2. —64 3. 56 4. —21 5. 83

28 —39 38 —]7 75

6. 13$ 7. 256 8. —384 9. 489 10. 341

96 209 —I23 261 96

11. —482 12. 362 13. —416 14. —3684 15. —4829

— 82 94 — 329 — 2318 — 3264

16. 4,826,378 17. 3,821,987 18. 2,381,236 19. —3,472,$31

2,918,436 2,943,998 493,648 —2,574,283

20. 5,000,000 21. —4,000,000 22. 3,000,000 23. 6,000,000

4,872,381 — 278,923 2,872,917 378,293

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MAT 082 Basic Arithmetic Page 14 of 105 Spring 20] 0

1.5 Basic Ope~'ations: Subtraction This is adding numbers with different signs.

A plus sign (+) means positive or plus. A negative (--)sign means minus or negative.

E-- left is the negative direction

Adcliug oxx the number dine:

right is the positive direction—~

'his is adding numbers which have the same sign.

Consider +3+5 . On the number line you start at 0 then go to the right 3 spaces, then go to the

right 5 more spaces, This puts you at -+-8.

Consider -2-4. On the number line you start at 0 then go to the left 2 spaces, then go to the

left 4 more spaces. This puts you at -6.

Adding numbers with opposite signs is called subtraction.

Subtraction is the opposite operation of addition. This is why the "+" direction is to the right,

and the "-" direction is the opposite way, to the left.

Consider +8 --5. On the number line you start at 0 then go to the right 8 spaces, then go the

left 5 spaces. This puts you at +3.

Examples: Add the following numbers.

+15-10 -15+10 +17-8 -~17+8

Examples with larger numbers: Add the following numbers.

+sl +189 +5728

—21 —122 —2345

+500— 200

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15 Basic Operations: Subtraction Homework

1. +41 2. +64 3. —56 4. +21 5. +$3

—28 —39 +38 —17 —75

6. 13$ 7. 256 8. + 384 9. + 89 10. + 341

— 96 — 209 —123 — 261 -- 96

11} 482-82 12) —362+94 13) —416+329 14) +3684-2318 15) —4829+3264

16. + 4,826,378 17. —3,821,987

—2,918,436 + 2,943,998

20. + 5,000,000 21. 4,000,000

—4 872,381 —278,923

1 g. +2,381,236

— 493648

22. --3,000,000

+ 2,872,917

19. —3,472,831

+ 2,574,283

23, +6,000,OQO

— 378,293

24. Mario paid $885 for his motorcycle while Bob paid $1348 for his. How much more did Bob

pa y than Mario for his motorcycle?

25. Joaquin shot a 132 at the golf course last week while his wife shot an 87. What was the

differen ce in their scores.

26. Maria bowled a 207 last evening in the Bowling 'I'ouznament. The. tournament was won

with a score o f 281. What was the difference between Maria's score and the wirmix~g score?

27. Carlos was looking at a boat which cast $1595. He was told by the store manager that he

could save $475 b y waiting until the boat was on sale. 7f Carlos waited for the sale, how

much would the boat cos t him then?

28. Mrs. Trujillo placed a meat order for 300 pounds of beef. When she went to pick it up she

was told she could onl y have 175 pounds. How much difference was there between what

Mrs. Trujillo wanted and what she achxally received?

29. Jim made a 7$ on his first math exam while Francisco made a 42. How many points better

did Francisco do than Jim?

30. Amalia lived 438 miles from Denver while Veronica lived only 289 miles. How much closer

did Veronica lave to D enver than Amalia?

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1 6 Basic Operations: Multiplication.

MULTIPLICAND: The number that is being multiplied.

MULTIPLIER: The number doing the multiplying.

PRODUCT: The answer we get in a multiplication.

FACTORS: The numbers being multiplied together (the multiplicand and the multiplier).

6 MLTLTIPICAND 6 x 8 = 6 • 8 = (6) (8) = 48

x 8 Ivl [JLTIPLIER

4$ PRODUCT

Multiplication is indicated by using parentheses ( ) ( ) or a dot "•". Here is an example of the

dot between the numbers 4 • 3 and the use of parenthesis (4} (3).

Multiplication is really repeated addition. That is, (6) (3) is really 3 + 3 + 3 -I- 3 + 3 + 3 = l 8.

.M.~~C.~~ ~.~g~o~:m-,~.1~s ~~

1 2 3 4 5 6 7 8 9 10 11 12 13

1 1 2 3 4 5 6 7 8 9 10 11 12 13

2 2 4 6 8 10 12 14 16 18 20 22 24 26

3 3 6 9 12 15 18 21 24 27 30 33 36 39

4 4 8 12 16 20 24 28 32 36 40 44 48 52

5 5 10 15 20 25 30 35 40 45 50 55 60 65

6 6 12 18 24 30 36 42 4$ 54 60 66 72 78

7 7 14 Z1 28 35 42 49 56 63 70 77 84 91

S 8 16 24 32 40 48 56 64 72 8Q 88 96 104

9 9 18 27 36 45 54 63 72 81 90 99 108 117

10 10 20 30 40 50 60 70 80 90 100 110 120 130

11 11 22 33 44 55 66 77 $$ 99 110 121 132 143

12 12 24 36 48 60 72 84 96 108 120 132 144 156

13 13 26 39 52 65 78 91 104 117 130 143 156 1b9

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1.6 Multiplication Homework

Multiply

1. 38 2. 2$ 3. 22 4. 73 5. 68 6. 49

2 6 7 6 7 8

7. 32(9) 8. 18(8) 9. 35(5) 10. 42(8) 11. 39(4) 12. 28(3)

13. 29.7 14. 82.9 15. 69.8 16. 30.6 17. 7842 18. 39.21

Multiply

19. 62 20. 21 21. 37 22. 28 23. 21 24. 37

42 78 48 73 98 43

25. 31 26.723 27.138

8 4 6

31. 168 32. 287 33. 361

12 32 212

37. 382(726) 38. 398(219)

28.213 29.315 30.142

18 13 18

34. 203 35. 368 36. 368

142 102 213

39. 364(13} 40. 3462(213)

41. Ernesto went to the market to buy steaks which cost 3 dollars per pound. When he had

the steak weighed, it was 12 pounds. How much did the steaks cost?

42. Gonzalez Canr~iiig purchased 23 canning machines which cost $1683 each. How

much did they pay for all of the machines?

43. The Buena Vista Hotel has 412 rooms. If each room rented for $14 per night, haw

much would the hotel make if they had a night when all the rooms were filled?

44. The Bravo Carpet Company would like to carpet your entire house at a cost of

7 dollars per square yard. It was found that your house contained 138 square yards.

how much would you have to pay to have the carpeting done?

45. Ofelia's Catering Company is planning a paving project in your neighborhood. The county

says it will cost each family $185. There are 175 families in your neighborhood.

how much will the project cost?

46. Ana Lisa bought 498 dresses wholesale for her dress shop. The company charged her

12 dollars per dress. How much did Ana Lisa pay for the dresses?

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1.7 Long_,Division

Zero divided by any number is zero.

~ = 0 because 8(0) = 0 0 =132 = 0 ~ = 0 9 = 0

8

zero divided by 8 is 0

Division by zero is undefined N g 132=0 12 g

0 0 0 0

These are all called undefined quantities.

QUOTIENT:_ The answer to a division problem.

DIVISOR: The number doing the dividing. DIVIDEND: The number that is being divided.

Division can be indicated in three different ways.

This is called

Long Diviszon

4 quotient Using the

divisor 3~ 12 dividend Division Symbol

12 12 — 3 = 4

dividend divisor quotient

Fraction form

dividend

12

— 4 quotient

3T

divisor

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Steps for Long Division

342 ~ 12

The division algorithm

1st. Divide

2nd Multiply

3`d Change the sign

4~' Brrng down the next term

Repeat the steps

1. Set up the problem so that 12 is outside and 342 is inside.

2. Does 12 divide into 3? No.

3. Does 12 divide into 34? Yes, 2 times.

Ptace the 2 above the 4.

4. Multiply 2 x 12 = 24 and place the product 24 below 34.

5. Change the si,~n of 24 to minus 24. Add +34 and —24.

6. Bring down the 2 in the dividend to become the 102.

7. How many times will 12 divide into 102?

(8 times since 8x12 = 96.)

8. Place the 8 above the 2 in the divisor.

9. MultipCy the 8 times the 12 and place the product

raider the 102.

10. Change the sign of the 96 to minus 96. Subtract

12) 342

2

12)3 4 2

24

The division algorithm:

1st. Dfvide

2nd Multiply

3ra Change the sign

4`' } Bring down next term

Repeat the steps

Since there is no ntunber to bring down and 12 will not divide into 6,

we have a remainder of 6. Our final problem looks like this.

or 28 6 or

l2

The main thing to keep in mind here is to continue the dividing and subtracting until you can't

divide any longez and there is nothing else to bring down.

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1.7 Long Division Homework

Divide from memory or use Iong division.

1. 32=8 2. 24=4 3. 36:-6 4.54=9

5.728 6. 648 7. 426 8. 63-9

9. 15=5 10. 81=9 11. 922 12. 427

13. 4207 14.0:-9 15.121=I1 16.144=12

17. 3 = 0 18. 400 ~ 20 19. 800 = 25 20. 650 ~ 13

21. 387 = 8 22. 492 ~ 9 23. 682 =11 24. 893 =12

25. 312 =13 26. 18.72 = 9 27. 689 = 42 28. 93 8:81

29: 298 =13 30. 1647 = 3 31. 8461= 36 32.8136 ~ 142

33. 436 =12 34. 893 ~ 11 35. 3846 ~ 132 36. 847 -: 12

37. 693 :-9 38. 1382 :-12 39. 841=89 40. 4632=13

41. Three men received $210 for a job they did together. If they each received equal amounts,

how much did each znan receive for the work done?

42. Ajax Car Rental purchased 38 cars for a total of $144,400 if all of the cars were the same

price, how much did each car cost?

43. 16 baseball gloves of the same price cost a total of $384. Haw much did each cost?

44. Six bowlers had a combined score of 1248. You know that they all bowled the same score.

What did each Bowler Bowl?

45. 25 restaurants in Bo~:ville contributed $625 to the restaurant association. Assuming they all

contributed the same amount, how much did each contribute?

46. Mary Ann bought enough material for 6 bridesmaids dresses. Her total bill came to $138.

What price would she ha ve to charge for each dress?

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1.8 Divisibility Tests

Divisibility: A whole number is divisible by another whole number if the remainder is 0.

There are some quick test you can use to decide whether one number is divisible by another.

Divisibility by 2: A number is divisible by 2 if the number ends in 0, 2, 4, 6, or 8

Example 1:

Are all the following numbers divisible by 2?

(a) 986 is divisible by 2 because the number 986 ends in 6.

(b) 3255 is not divisible by 2 because 3255 does not end in 0, 2, 4, 6, or 8.

Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.

Example 2: Are all the following numbers divisible by 3?

(a) 4251

Add the digits: 4 + 2 + S -f- 1 = 12 Since 12 is divisible by 3, the ntunber 4251 is divisible by 3

(b) 29,806

Add the digits: 2 + 9 + 8 + 0 + 6 = 2S Since 25 is nat divisible by 3, the number 29,806 is also

not divisible by 3.

CAUTION

Be careful when testing for divisibility by adding the digits. This method works only when

testing for divisibility by 3 or by 9.

Divisibility by 5 and by 10.

A number is divisible by 5 if it ends in 0 or S A number is divisible by 10 if it ends in 0

Exaniple 3: Are the following numbers divisible by 5?

(a) 12,900 ends in 0, so it is divisible by 5.

(b) 4325 ends in 5, so it is divisible by 5

(c} 392 ends in 2, so it isn't divisible by 5.

Example 4: Are the following numbers divisible by 10?

(a) 700 and 9140 end in 0, so both numbers are divisible by ~0.

(b) 35S and 18,743 do not end in 0, so these numbers aren't divisible by 10.

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Test for Divisibility

A number is divisible by 2 if it ends in 0, 2, 4, 6, ax 8.

A number is divisible by 3 if the sum of its digits is divisible by 3.

A number is divisible by 4 if the last two digits make a number that is divisible by 4

A number is divisible by 5 if it ends in 0 or 5.

A number is divisible by 6 if it is divisible by both 2 and 3.

A number is divisible by S if the last three digits make a number that is divisible by &.

A number is divisible by 9 if the sum of its digits is divisible by 9.

A number is divisible by 10 if it ends in 0.

1.8 Divisibilifiy Tesfs Homewoxk

Determine if the following 28 numbers are divisible by 2, 3, 5, or 10. Put a ✓mark in the

blank if the number is divisible by the number at the top of each column. Use the

divisibility tests and explain your answers. Number 1 is done for you below.

2 3 S 10 2 3 5 10

1. 30 ✓ ✓ ✓ ✓ 15. 8302.

2. 184 16: ; , `

32,472

3. 445 17. 612

4. 903 1g• 315

5. 5,166 19. 2,714

6. 21,763 20. 36,000

7. 25 2~. 836

8. ..192 22. 7,545

9: 897 23. 242,913

10. 500 24. 102,484

11. 160 25. 290

12. 635 26• 218

13. 3,381 27.. 2,020

14. 108,605 28. 11,670

I. 30 is divisible by 2, it ends in 0; it's divisible by 3 because 3+0 is divisible by 3;

30 is divisible by 5 because it ends in 0; 30 is divisible by 10 because it ends in 0.

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1.9 Checking Division

quotient+remainder 4R5

divisor- dividend 6 29 6xG = 24 24+5 = 29 ~

Multiply the divisor by the quotient and then add the remainder. The result is the dividend.

Example 1; Check each quotient. If the quotient is wrong, re-do the division correctly.

91R3

(a) 5 458

1.9 Checking Division Homework

Divide and Check

29R4

(b) 6 1258

1. 2 225 2.3 275 3. 4 538 4. 5g 9 5. 5 937 6. ~6 5 7. 3 1885 8. 8 1135

Check each division. Tf a quotient is incorrect, fmd the correct quotient.

38R1 92R2 328 46R3

9. 3 115 10.8 743 I1. 4 1312 12.5 2033

#13-22 Divide &rewrite each problem using two other division symbols: =or j or fraction bar.

13. ~~ 14, 9 15. 24=0 16. 4~-4 17. ~

12 0 4

18.0:-8 19. 0=12 20. ~ 21.0 21 22. 2=0

#23-26 Divide and identify the dividend, the divisor, and the quotient.

23.4108 24.1355 25.324=9 26.8176 27.69137

28.9 8371

33. 15018=3

29.6 1854 30.8 856 31. 4024 -: 4 32.16, 024 = 8

34.320Q8=8 3S. 

26,684 36. 16,398 37. 74,751

38. 72,543 3~

5

Check each quotient

67R2

X33.7 4692

3, 003R5

47.6 18, 023

4 9 6

71, 776 40. 77, 621 41. 128, 645 

42. '

255

7 3 7 4

If a quotient is incorrect, divide again to find the correct quotient.

663R5 3568R2 25, gay

44.9 5974 45, 6 21, 409 46. 4103, 516

4,20$ 11,523R2 27,532R1

48.8 33, 664 49.6 69,140 50.3 82, 598

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1.10 Exponents

The exponent tells us how many times to repeat multiplication. -~ Z 3 is 2 • 2 • 2

215 =2.2.2.2•,..•2 On the calculatoruse2n15

w

15 timcs

2 is called the base

l S is called the exponent or power

base 

]'32 ~—~xponenl~ 3 to the 5eCQ11CI ~OWeI ~--> .3~3~ —~ 9

24 is read as 2 to the fourth power which means 2(2)(2)(2} this is 16.

Examples: S

63

6.6.6

z~~

X45)

45

implify the exponential expressio

( lz

\5/

~s~~s~

9

zs

z.

~2~4

5

~3~

36

Examples of more complicated exponential expressions: Notice the characteristics.

There are two exponential expressions. They are being multiplied. They have the same base.

6z• 65 34. 3Z

6.6. 6.6.6.6.6

6~

279,936

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( a 3~ z5

l32 ~ 35 7 ~

~32~~32~~32~~3~~

~3.3~~3.3~~3.3~~3.3~

3.3.3.3.3.3.3.3

3g

6,651

a° =1 ,note : a ~ 0 Something raised to the zero power is usually "1"

Examples: Simplify the expression.

5° is 1 ~17~~ is 4°

3~

51° 2° +3° 

~8J

1.10 Egaonents Homework

Simplify the expression.

z 3 54

1. 23 2. ~23~ 3. ~Sz ~ 4. Z

5

a z / Z ~2 3

5. ~3~ 6. ~~~ 7. 35 .32 8. `~ 2

B

9. 4' 10. (23)(2") 11. 66 12. (3°)(5°)

4 p ,~6 2

i3. 1 5~ 14. 7s 15. ~9~z~ 34 16. ~3Z)~3)n ~g~)

MCCCD/Martinez00556

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1.11 Oder of O~eratians

The four basic operations are &

If an expression has all four operations, in what order should the operations be performed?

Example: Simplify 4+ I 8 :- 2 —1 • 5 -t- 2

Addition: a+b ~ basic counting concept

Subtraction: a — b = a + (—b) ~ the opposite of addition

Multiplication: -~ 3(2) is 2 + 2 + 2 ~ repeated addition

9.4 is 4+4+...+4

Y

9 4's being addcd

Division: a = b = ~ = a • ~ ~ the inverse of multiplication

Exponents (Powers): 4' = 4~4~.. 4 ~ repeated multiplication of the base

7 timcc

Example 1 "Multiplication is repeated addition"

3.44+4+412

4.6~

2 • (-7) ~

Example 2 "Exponents indicate reheated multiplication"

34 ~3.3.3.3~81

(5)3 ~ (5) ~ (5) • (5) ~ 125

(7)" ~

2~0 ~

Example 3 "Exponents indicate repeated

52 exponent (power)

5 ~ 5 multiplication

5+ 5 -F 5+ 5+ 5 addition

25 calculation

3'

Example 4 Follow example 3. Notice the order of operations in the example.

43 exponent

multiplication

addition

final value

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Example 5 Simplify the following expressions.

4.3Z +5.7 32 +4z 4.52 -3.24

2+4.3~+5.7 4+3Z +42

What if we want to add before multiplying? Or multiply before the exponent?

Answer: Put the part you want to do first in parentheses.

Grouping is indicated with parentheses ( ), with brackets [ ], and with braces { }.

If an expression has all four operations and also has exponents and grouping symbols, in what

order should the 6 operations be performed?

Example 6 Simplify the following four expressions

(2+3)z 5•(3-23) 3•(5-3)z 1+2(34 -16)+15-24-3 • 6+2

Grouping is also indicated by the traction bar. For example, to simplify the expression

13+5-7+1

+ 9 you would first need to simplify the group of numbers in the numerator and

8-3-1

the group of numbers in the denominator.

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Steps for Order of Operations

1. Crrouping: Parentheses () Brackets [ J Braces { } numerator and denominator

2. Exponents (Powers)

3. Multiply or Divide in order of occurrence from left to right

4. Addition e~-tr~t~ae~~

Example 7

32+4z 12(1+3)~6+8-2 8(1+3)=2+(3+4).-7-8

7-2

1.11 Order of Oaerations ~-Iomework

Simplify the following expressions.

1. 3.4+5.2 . 2. 5.3~-2.7 3. 4(3+2)2

4. (5--3)z •(2+1)z 5. 5•(3+7}-2~6 6. (3.4--5)3

7. (14-6)2 -+-(5--3)3 8. 5•(12-8)+3•(2+4) 9. S+3.24

10. (3-1)'+(1+2)S 11.

i z 12 +5 12 ~ z 6- -5

15-2 22

13. 8+4.2-3~4 14. 3 ~5~ -~z~ 1S. 4.35 -3.2'

(S - 2}z

16. 3+2.15 17. 5•(13-7)Z +2•(S-2)' 18.

3

5 +~

a

19. 3 (4 5~> 

20. (13 - 5)' ~ (2 + 3}~ 21. 23 - 2 ~ (14 - 9~

22. Sz +2.5-8 23. -34.5+(5)2.3 24. 5.4-32

25. 2 ~ (3 + 4}3 26. (3)4 + (5}~ 27. -34 + (5)'

28. (3)'-Sz 29. (3.8+Sz)-(3~-4~2) ~0. 8.3-5.2

SZ -3'

MCCCD/Martinez00559

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7..12 Mean, Median and Mode are three statistical measures.

The mean is the average and is found by first adding all values to get the total then

dividing your answer by the number of values. mean = 

add all numbers

number of values

Example 1: Find the mean (or average) of these scores: 84, 90, 95, 98 and 88

The mean = 

g4+ 90 + 95 + 98 + 88 _ 455 _ 91

5 5

The mean score is 91.

Example 2: The sales of photo albums at Sarah's Card Shop for each day Iast 'week

were $86, $149, $103, $118, $117, $126, $158 and $149. Find the mean sales.

Answer: T'he mean sales amount is

The median is the middle number in a list of numbers when listed in order from

smallest to largest.

Steps to finding the Median of a set of numbers

1. Arrange the numbers from smallest to largest.

2. Lf there is an odd number of values, the middle value is the median.

If there is an even number of values, the average of the two middle_ values is the

median.

Example 1: Find the median of these scores: 84, 90, 95, 98 and 88

Answer: The median score is

Example 2: The sales of photo albums at juanita's Card Shop for each day last week

were $86, $103, $149, $118, $117, $126, $158 and $149. Find the median for the

sales of photo albums.

Answer: The median score is

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29. A car travels 220 miles in 4 hours. What's the rate of the car in miles per hour?

30. A train travels 360 miles in 5 hours. What's the rate of the train in miles per hour?

31. The flow of water from a water faucet can fill a 3-gallon container in 15 seconds. Give the

ratio of gallons to seconds as a rate in gallons per second.

32. A car travels 95 miles on 5 gallons of gas. Give the ratio of nniles to gallons as a rate in miles

per gallon.

Twenty-one people work in an office which has 15 desks and nine computers.

33. What is the ratio of desks to people?

34. What is the ratio of computers to people?

35. The ratio of females to males is 6 to 7. The class has 39 students. How many females are

there?

36. The ratio of females to males is 6 to 7. The class has 39 students. Haw many males are

there?

MCCCD/Martinez00561

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•

Ntahatv~.a ciawdh~

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3 GEOMETRY

3.1 Pointsx Lines, Planes, &Ankles Draw each of the following representations.

Point Dot, no length, width or thickness

Line Goes through two points, extends infinitely in both directions (A line is an infinite number of poinu.)

Line AB Half Line AB Ray AB Line Segcnenf AB

Plane A flat surface with no thickness & no boundaries

A ra is a set of points starting at a point and continuing in one direction forever.

Note: A ray can rotate 360° or more

Ankle Formed by two rays starting at the same point

.~At3C = 1CCBA

Acute angles are between 0 degrees and 90 degrees. They are sharp (i.e. pointy) angles.

A right angle measures 40 degrees. The rays are perpendicular 1.

Obtuse angles are between 90 degrees and 180 degrees. They are dull angles.

Straight angles measure 180 degrees. They are lines.

Reflexive angles are between 180 degrees and 360 degrees.

B

C

A 0

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Example 1. Draw and describe a 30 degree angle.

-----~

Answer: A 30 degree angle is 1/6 of a straight angle. It is an acute angle.

Example 2. Draw and describe a 120 degree angle.

Example 3. Draw and describe each of the following angles.

zero

degrees 60 degrees 90 degrees 180 degrees 360 degrees

right angle straight angle reflexive angle

Similar ~'i~ures are figures that have the same shape but not necessarily the same size.

Draw two similar rectangles with corresponding sides and corresponding angles.

Complementary Andes are two angles whose measures add up to 90°.

Draw two complementary angles.

Supplementary Andes are two angles whose measures add up to 180°.

Draw two supplementary angles.

Parallel Lines are lines that have the same slope and never intersect.

Draw two parallel lines.

Intersecting Lines are tines that intersect at one point.

Draw two intersecting lines which form 20 degrees.

Perpendicular Lrnes are lines that intersect at a 90° angle.

Draw two perpendicular lines.

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3.1 Andes Homework Draw &describe the angle: acute, right, obtuse, straight, reflexive

Zero degree angle 2. 30 degree angle. 3. 45 degree angle 4. 60 degrees

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3.2 Two-llimetesional Figures and their characteristics

Always draw the geometric frgure and label the necessary features before doing any work.

Triangles 10.2 Veronica's

A Triangle is a geometric figure of 3 sides

* The sum of the measures of the 3 angles of any triangle is l 80"

Types of Triangles (pictures)

Acute Right Obtuse

All angles are acute One right angle One angle is obtuse

Isosceles Equilateral Scalene

Two side have equal length all side axe equal no two sides are equal

Trian~es have three sides.

The three angles of any triangle add up to 180 degrees.

The base is the horizontal segment that the triangle sits on.

The heightrs vertical segment from the top of the triangle to the base.

The height of a triangle is perpendicular to the base of the triangle.

I) 2 sides are the same length and 2 angles have the same number of degrees =Isosceles

2) Has 2 legs and a hypotenuse; one angle is 90 ° ; az + bZ = c2 Right Triangle

3) All sides are different lengths and alI the angles have different number of degrees =Scalene

4) All sides are the same length and alI angles are 60 degrees= Equilateral

Trianele shaves and names:

Isosceles Triangle Right Triangle Scalene Tri¢ngle Equilateral Tn:¢ngle

aZ +b Z =c'

2 sides are the same length The Pythagorean Theorem All sides are different lengths All sides are

the same length

2 angles have the same I-Cas 2 legs and a hypotenuse All the angles have different number All angles are

60 degrees

number of degrees One angle is 90 degrees of degrees

~~

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Quadrilateral t~ = a 4-sided geometric figure.

Parallelogram ~ 2 pairs of sides are parallel.

Rectangle ~ 4 right angles.

Rhombus ~ 4 equal sides.

Square ~ 4 equal sides and 4 right angles.

Trapezoid ~ 1 pair of parallel sides.

Quadrilaterals have sides and angles.

The four angles add up to degrees

The height is perpendicular 1 to the base.

Quadrilaterals are made up of two triangles.

Parallelograms have 2 pairs of parallel sides, Parallelogram ~ 2 pairs of sides

are parallel.

The base and the top are the same

The left side and the right side are the same

The height is perpendiculaz 1 to the base.

The height does not equal the length of left or right side.

Is a parallelogram a rectangle?

Answer:

Rectangles have four angles.

angles.

Kectangles have two pairs of sides.

A rectangle is made up of triangles.

Adjacent sides of a rectangle are

The base and the top are the same

The left side and the right side are the same

71~e base is sometimes called the

The side is sometimes called the height or the _

Ys a rectangle a parallelogram? Answer:

Is a rectangle a square? Answer:

Rectangle ~ 4 right

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Rhombus t~ 4 equal sides. Draw a rhombus.

A rhombus has

Is a rhombus a square?

equal sides. This is like a slanted square or a regular square.

Answer:

Square a 4 equal sides and 4 right angles.

The sum of the angles =

A square has four equal sides and four z~ght angles. Is a square a rhombus?

Answer:

Is a square a rectangle? Answer:

Trapezoids have one pair of parallel sides.

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Circle

r~rcun~fere;~~a

1. A circle is made up of points which are all the same distaance from a point called the center.

2. The diameter is a set of points starting on the circle going through the center to the other

side of the circle.

3. The secant is the set of points starting on the circle not through the center to the other side of

the circle.

4. The radius is the set of points from the center to the circle.

5. The circumference is the distance around the circle. This is the perimeter of the circle.

6. A circte has 360 degrees.

3 2Two-Dimensional Figures Homework

Draw each of the following and describe the characteristics of each.

1. Isosceles Triangle

4. Equilateral Triangle

7. Reflexive angle

10. Quadz-ilateral

13. Rhombus

16. Circle

19. secant

2. Right'I'riangle

S. Acute angle

8. Straight angle

11. Parallelogram

14. Square

17. diameter

2U. Circumference

3. Scalene Triangle

6. Obtuse angle

9. Right angle

12. Rectangle

15. Trapezoid

18. radius

21. Draw a triangle. How many degrees are there in a triangle? Use complete sentences.

22. Draw a rectangle. How many degrees are there in a rectangle?

23. Draw a quadrilateral. How many degrees are there in a quadrilateral?

24. Draw a circle. How many degrees are there in a circle?

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3.3 The Pytha~arean Theorem c

a

b

a` . b` . 

~:

The two sides which make the right angle are called the legs and the lengths are zepresented by a

and b. 'I1~e last side is the longest side and it is called the hypotenuse. Its length is represented

by the letter c.

Pythagorean Theorem: CIz -F- b2 = CZ Add the squares of the legs. This equals the hypotenuse squared.

In a right triangle, the square o£ the hypotenuse is equal to the sum of the squares of the other two sides.

/f

4

~~ G

3 1

9~t-16-25 1+4=5

Exannple 1. Draw a right triangle where a is 6 inches and b is 8 inches. Find the length of c.

Example 2. A wire from the top of a 12-foot pole is fastened to the ground 5 feet from the

bottom of the pole. What is the length of the wire?

Example 3. Draw a right triangle where a = 12 cm and c = 20 cm. Find the length of b.

Example 4. Draw a right triangle where a = 7 ft and c = 25 ft. Find the length of b.

Example 5. Draw a right triangle where b = 5 in. and a = 4 in. Find the length of c.

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3.3 The Pythagorean Theorem Homework

llraw each triangle. Find the length of the missing side.

If the answer is not a whole number, then give the actual answer and the answer to two (2)

decimal places.

1. a=5 yd. and b= 5 yd. 2. a=15 in., b=9 in. 3. a=6 ft, b=Eft. 4, a=18 m, c= 30 m.

5. A wire from the top of a 24-foot pole is fastened to the ground 10 feet from the bottom of the

pole. How long is the wire?

6. A Iadder is leaning against the top of a 15-foot wall. If the bottom of the ladder is 20 feet

from the wall, how long is the ladder?

7, a=5 ft., b= 2 ft.

8. a=5, b=7

9. The width of a television screen is measured by the length of the diagonal. If the diagonal is

27 inches, and the height is 17 inches, fmd the length of the television.

10. Is a triangle with sides of 5, 12, and 13 meters a right triangle?

1 I . Ys a triangle with sides of 2, 2, and 3 feet a right triangle?

Find the missing side.

12. a=15, c=17. 13. b=45, c=53. 14. a=5, b=7. 15. a=1, b=7.

16. The lengths of the three sides of a triangle are $, 15, and 17. Determine whether the triangle

zs a right triangle.

17. The lengths of the three sides of a triangle are 7, 24, and 26. Determine whether the triangle

is a right triangle.

18. The lengths of the three sides of a triangle are 6, 8, and 10. Determine whether the triangle

is a right triangle.

19. The lengths of the three sides of a triangle are 9, 39, and 40. Determine whether the triangle

is a right triangle.

20. A tree casts a shadow 24 feet long and a man 6 feet tall casts a shadow 4 feet long. Find the

height of the tree.

21. The gable end of the roof is divided in half by a vertical brace which is 8 feet tall. The base

is 30 feet. Find the length of one side of the roof line.

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22. Find the length of the hypotenuse of the right triangle whose legs measure 5 feet and 12 feet.

Draw and label the figure.

23. A 45-foot tall ladder is fifteen feet from the base of a house. How tall is the house? Draw

and label the figure.

24. Find the length of the missing leg of the right triangle when the hypotenuse is equal to 60

inches and one leg is 40 inches. Draw and label the figure.

25. To get from point A to point B, you must avoid walking in a pond. To avoid the pond, you

must walk ] 8 yards south and 6 yards east. If you had on the proper shoes and could walk

through the pond, how many yards would you have walked? Draw and label the figure.

26. Thumbelina is 10 centuneters tall. She casts a 15 centimeter shadow. What is the distance

from the top of her head to the tip of hex shadow? Draw and label the figure.

27. To prove a point, architect Carlos built a pole that casts a 14-foot shadow. The distance

from the top of the pole to the edge of the shadow is 36 feet. How ta11 is the pole? Draw and

label the figure.

2$. A suitcase is 12 inches high and the diagonal distance across it is 48 inches. What is the

width of the suitcase? Draw and label the figure.

29. A 1 S-foot house has a 17 foot cable line stretched from the roof. What is the distance from

the base of the house to the end of the cable wire? Draw and Iabel the figure.

30. Olivia and Andrea start out at the same point. Olivia travels 8 yards south and Andrea

travels 22 yards southeast. What is their final distance away from each other? Draw and label

the figure.

31. Two sprinters run 3 feet south and 4 feet east. What is the shortest distance you have to

travel to reach their starting point? Draw and label the figure.

32. The Martinez family plans to carpet their new family room which is 30 feet long and 21 feet

wide. TYze carpeting costs $13 per square yard. How many square yards will the family need

and what will be the total cost for the carpeting?

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3.4 Three-]dimensional Figures

A rectangular prism is called a rectangular box or a rectangular base solid.

Tt is called a rectangular prism because the base is a

It has 12 edges. Number them 1 to 12 in the diagram. ,

w ,, »»>> ~~ h area

A cube has all its sides the same length.

The base is a

How many edges does it have?

Answer: The cube has edges.

■

A circular prism is the shape of a can. This is also called a circular base solid or a

cylinder.

It is called a circular prism because the base is a

If you anroll the can, you will have two

and a

A triangular prism is the shape of a _

The base is the shape of a

If you laid it out flat, you would have two

anda

A s here is the shape of a ball.

It is made up of points which are equidistant from a point called

the center; it has a radius and a diameter.

h~

z•

H h

cra a: ~ .. b

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3.4 Three-Dimensional Figures Homework

Label the parts ~ for length, uc for width, r for radius, d =diameter, fi. for height, ~. =base,

c for center.

1. Draw two different rectangular prisms and label the parts

2. Draw two different circular prisms and label its parts.

3. Draw two different triangular prisms and label its parts.

4. Draw a sphere and label its parts.

Write a complete sentence to answer the following questions.

S. How many sides does a rectangular prism have? Draw one and number the sides.

6. How many sides does a cube have? Draw one and number the sides.

7. Haw many sides does a triangular prism have? Draw one and ntunber the sides.

8. How many corners does a rectangular prism have? Draw ane and number the corners.

9. How many corners does a cube have? Draw one and number the corners.

10. How many corners does a triangular prism have? Draw and number the corners.

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3.5 Perimeter

The perimeter is the distance around the rim of a figure. It is sometimes considered a fence.

The perirraeter is represented by the capital letter P.

Triangles and perimeter

For a triangle, the pezimeter is the sum of the lengths of the three sides.

Perimeter ~ P = a + b + c units

Example 1: Draw the triangle then find the perimeter of a triangle whose sides are 5", 12", and

3 ".

Answer: P = a + b + e Draw [he picture here

P = 5+12+3

P = 20 inches

Example 2: What is the perimeter of a triangle whose sides are each 13"? Draw the triangle.

Rectangles and perimeter

The perimeter of a rectangle is the sum of the lengths of the four sides.

P=b+b+h+h

P = 2b + 2h or Perimeter ~ P = 2l + 2w units

Example 3: What is the perimeter of a rectangle whose base is 15' and the height is 12'?

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Circles and perimeter

For a circle, the perimeter is called the circumference and is represented by a

capital C.

~r is an irrational number called "pi." It's value is approximately 3.14 or z2

7

P=C= n * d

Example 4: What is the circumference of a circle with a radius of Sxn? Grve exact answer and

answes rounded to one decimal place (where appropriate).

Rectangular prisms and perimeter

ZU . b

»»>~ 1z area

P = b+b+b+b-E-w+w+w+w+h+h+h+h

P = 4b + 4w + 4h units

Example 6: Find the perimeter of the rectangular prism where b = 6', w = 12', and h ` 12'.

Triangular prisms and perimeter

P=a+b+c-+-h+h+h+a+b+c

h P = Za + 2b + 2c + 3h units

,., Example 5: What is the perimeter of the triangular prism whose long edges measure

°"°' h 13" and the short edges measure 7", 3", and 6"? (Note: Be sure to draw and Jabel

the figure properly.)

Circular Cylinder or Circular Prism A circular prism is the shape of a can.

It is called a circular prism because the base is a circle.

If you unroll the can, you will have rivo circles & a rectangle.

P=mod +Ord +h+h

P = 2 ~cd + 2h ~-- ~ d

h

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3.5 Perimeter Homework

Give exact answer and answer rounded to one decimal place.

1. What is the distance around a figure called?

2. Draw each of the following figures and give the formula for the perimeter (or circumference).

Triangle Rectangle Circle Rectangulaz Prisrn

Triangular Prism Square Rhombus Parallelogram

3. What is the perimeter of a triangle whose sides measure 3.1 inches, 5.175 inches, & 7.35

inches? Give the answer in two forms: as a decimal and as a mixed number.

A circle has a radius of 5.2 meters. Use the decimal approximation of pi,

4. Find the diameter of the circle.

5. Find the circumference of the circle.

6. A triangle has sides which measure 4 1/3 yds, 8 3/8 yds, and 2 3/4 yds.

The height of the triangle is 3 yds. Find the perimeter.

7. Find the periz~z~eter of a rectangle whose sides measure 4 1/2 by 8 2/3 in.

8, Find the circumference of a circle with radius 5 feet.

9. Assume a right triangle has sides of 2 1/2", 3 1/2", and 5". What is the perimeter of the triangle?

10. Draw and label a circle with radius 12.7 feet. Give the formula for the circumference of the

circle.

11. Give the symbol for "pi" and two approxunate values of "pi."

12. Give the approximate value of the number "e."

13. Find the perimeter of the following figure.

4 5 in

4

16

11

14. Find perimeter of the following shape given that the length of b is 5 znches, the length of h is

3 inches, and the length of w is 17 inches.

wr~r»J 1 b I i area J

~1

I5. Find the perimeter of the following figure if the sides of the triangle are

4 meters, 3 meters, and 2 meters, and the height of the figure is 7 meters.

-'R?'gip i',b __

16. A rectangle has a length of 17.2 feet and a width of 6.9 feet. Find the perimeter.

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3.6 Area

The area is the number of squares in the interior of a figure.

2-dimensional areas and 3-dimentional areas

Area &Circumference 10.4 Veronica's

Formulas:

~ Area of a Rectangle: A= L x W (Picture)

• Area of a Square: A = Sz

• Area of a Parallelogram: A = bh

• Area of a Triangle: A = 2 bh

• A.rea of a Trapezoid: A = 2 h(a + b)

• Area of a Circle: A = ~rr~ ~r = 3.14 diameter = r + r = 2r

• Circumfrence —Distance around the circle: C = 2~'r A = ~trZ

Rectangles and area

Example l:

Consider a rectangle which has a base of 4 inches and a height of 3 inches.

How many squares are in the interior? Show the squares in the rectangle.

What is the area of the rectangle?

Answer: A = ~ square inches

height

or width

base or length

There are _squares in the interior of the rectangle.

The area of the rectangle is square feet (i.e. feet squared or

To find the area of a rectangle multiply the base times the liei~ht.

A=b*h or A=bh or A=1w

Area -~A = Iw square units

ft- ).

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Triangles and area

All triangles are half of a rectangle. Area ~,g = 1 bh - h b square units

2 2

(t.

Example 2:

Consider a triangle which has a base of 4 inches and a height of 3 inches.

How many squares are in the interzor? Show the squares in the triangle. This is not easy to do.

What is the area of the triangle?

Answer: A =

There are _squares in the interior of the rectangle.

The area of the rectangle is _square ft or ,ft squared.

To find the area of a triangle you can multiply the base times the height and divide by 2.

A = 1 b * h or A = l bh oz A = ! bh or A = l lw or A = bh or A = w I unitsz

2 2 Z 2 2 2

.~

~\\\~

Redraw each triangle so that one of the sides is horizontal. Thzs is called the base.

Put your pencil on the highest point on the triangle. Draw a line segment from this point to

the base.

Make sure the line segment is perpendicular 1 to the base. This is the height of the triangle.

Label the base and height of each triangle by writing "b" or "h" to identify the base and

height.

Example 1: Find the area of the triangle whose sides measure 7', 7', and 2'. The height of the

triangle is 6'.

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MAT 082 Basic Arithmetic

Circles and area

r~rcumferen~

Page 69 of 105 Spring 2010

Area --~ A = ~ ~ rz units Z

The area of a circle is 7l' times the radius squared.

Give exact answer and answer rounded to one decimal place

(where app~-oprrate).

Example 3:

Consider a circle with a radius of 5 cm.

How many squares are in the interior? Show the squares in the circle. "This is not easy to do.

What is the area of the circle?

Do this problem twice: once using 3.14 fox pi and the second time using ?; .

Answer: A =

Example 4:

Consider a circle with a diameter of 14 inches.

How many squares are in the interior? Show the squares in the circle. This is not easy to do.

What is the area of the circle?

Do this problem twice; once using 3.14 for pi and the second time using ~ .

Answer: A =

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3-Dimensional Areas

Area of a circular prism Cive exact answer and answer rounded to one decimal place (where appropriate).

Example 5:

Find the area of the circular prism whose radius is 3 inches and the height is 7 inches.

A=~crZ +~r~+doh

Area -~ A = 2nr Z + dh~c units ~

Area of a rectangular prism

Exampte 6:

Find the azea of the rectangular prism whose base is 3 feet, width is 4 feet, and height is 2 feet.

w »»l ~ b r2 ar~aJ A=bh. +b~c,~ +hw +hw +bw +bw ngM ead A erd from hxk ba~om r y

A = 2bh + 2hw + 2bw units2

3.6 Axea Homework

Give exact answer and answer rounded to one decimal place.

1. Draw each figure and give the formula for the area of each.

Triangle Rectangle Circle Rectangular Prism

Triangular Prism Square Rhombus Parallelogram

2. What is the area of a rectangle whose base is 8 2/5 inches and its height is three times

Phis number.? Give the answer in two forms: as a decimal and as a mixed number.

3. A circle has a radius of 5.2 meters. Use the decimal approximation of pi. Find the

area of the circle.

4. A triangle has sides which measure 41/3 yds, 8 3/8 yds, and 2 3/4 yds. The height

of the triangle is 3 yds. Pied the area.

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5. Assume a right triangle has sides of length 21/2", 31/2", 5". What's its area?

6. Find the area of the following shape.

16

11

S in

7. What is the area of a rectangle whose base is 8 2/5 inches and its height is three times

this nunnber? Give the answer in two forms: as a decirnaI and as a mixed number.

8. A triangle has sides which measure 41/3 yds, 8 3/S yds, and 2 3/4 yds. 'The height

of the triangle is 3 yds. Find the area of the triangle.

9. Find the area of a rectangle whose sides measure 41/2 by 8 2/3 in.

10. Find the area of a circle with a diameter of 10 feet.

11. Assume a right triangle has sides of 2 1/2", 3 1/2", and 5". What is the area of the

triangle?

12. Draw and label a circle with diameter 25.4 feet, then give the formula for the area of

the circle. Find the area.

13. A rectangle has a length of 17.2 feet and a width of 6.9 feet. Pind the area.

14. A circle has a radius of 7.5 inches. What is the diameter? Find the circumference and the

area.

Describe the puxpose of the following formulas. Draw and label the appropriate geometric

fz gore.

15. P=2b+2h

16. P=a+b+c

17. C=pitimesd

1.8. A=btimesh

19. A=1/2 btimesh

20. A = pi times rz

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3.7 Volume

A cube is a 3 dimensional figure whose sides are all the same size,

Volume tells how many cubes are in the interior of a 3~dimensional figure.

Volume 1 d.5 Veronica's

Rectangular Prism: V= L x W x H

Right Circular Cylinder: V = ~tr2h

Right Pyramid: V = 3 Bh , B =area of base

Right Circular Cone: V = 3 ~rrZh

Sphere: V = ~ nr3

To end the volume of a box, multiply the area of the base Yimes the height of the box.

~lJ ~ ~

,»» J ~2 area ~~ / ~ V = A ~ h

or V = b • h • w cubic units

V =1 ~w•h

Example 1:

Consider a box with a base of 3 inches, height of 2 inches, and a width of 14 inches.

How many cubes are in the interior? Show the cubes in the box. This is not easy to do.

What is the volume of the box?

Answer: A =

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The volume of a c~vlinder eauals the area of the base times the heieht of the cylinder.

Give exact answer and answer rounded to one decimal place (where

appropriate).

Volume

h~ V -A~h

V = TrrZ ~ h cubic units or units3

r

Example 2: Find the volume of the cylinder whose radius is 5' and height is 12'.

To end the volume of a triangular prismimultipl_y the area of the base times the hei~~tt.

Find the volume. b = 6m, h = 4m, H = l Om

h H

:: ~ i,.

area -1-

b

Example 3: Find the volume of a triangulax prism which is 20 inches tall, the b is 12 inches, and

the height of the base is 5 inches.

To fluid the volu~ue of a sphere multiply 4/3 times 1tY'2 . Give exact answers and answers

rounded to one decimdl place.

V = 4 ~r3 cubic units or units3 Area = 4 ~crz

3

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Example 4: Find the volume of the sphere with a radius of 6 yards.

Example 5: Find the volume of the sphere with a diameter of 10 cm.

3.7 Volume Homework Draw and label each geometric figure. Give exact answers and

also answers rounded to- one decimal place.

1. How many cubic feet of water are needed to fill a spherical water tank with a radius of 1S

feet?

2. Find the volume of the circular cylinder with diameter 6 cm and height l Ocm.

3. Find the voluzn:e of a rectangular solid with dimensions of 3 by 4 by S centimeters.

4. Find the volume of a rectangular solid with dimensions of 5 by 8 by 10 meters.

5. Find the volume of a prism whose base is a right triangle with legs 3 and 4 meters long and

whose height is 8 nneters.

6. Find the volume of a prism whose base is a right triangle with legs 5 and 12 feet long and

whose height is 10 feet.

7, Find the volume of a sphere with a radius of 9 inches.

S, Find the volume of a sphere with a diameter of 10 feet.

9. Find the volume of a cylinder with a height of 12 meters and a circular base with a radius of 6

meters.

10. Find the volume of a cylinder with a height of 4 meters and a circular base with a diameter

of 18 meters.

11. A classroom is 40 feet long, 3d feet wide, and 9 feet high. Find the number of cubic feet of

air in the room.

12. How many cubic feet are there in a cubic yard?

13. How many cubic inches are there in a cubic foot?

14. The largest refrigerator in New Mexico has a capacity of 25.2 cubic feet. How many cubic

inches is this?

15. The lifting power of a spherical balloon depends on its volume. How many cubic feet of gas

will a balloon hold if it is 40 feet in diameter?

16. A box of cereal measures 3 by 8 by 10 inches. The manufacturer plans to market a smaller

box that measures 2 z by 7 by 8 inches. By how much will the volume be reduced?

17. What units are used to measure area and what units are used to measure volume?

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MAT 082 Basic Arithmetic Page 75 of 105 Spring 2010

D D ~

a w~~~ ~.a~t~~,a vvo v~tia ~,

w~~~ t ~ r~~c ~, ~ ss ~ ~~

Y~ ~n,c s w ~c~ w~. ~

o ~ w ~l~ a ~, ~, o, ~~ ~a c a

~t-~ ~ ~ c o ~, ~ ~ ~. s ~ ~~, ~ ~ a v~,

a w1~ ~t ~ v~tia ~ ~ wl~r~

l~ ~r s w -~ I,~U ~ ~!~ at l,~ ~

,~4 W~S~ rD~e wc,DC~~~: ~sow~a ~sotow~.a~or

b~cowt.es f ~ ~rst ~at~wa ~s u.~prewc.e Court

J~cst~c~, ~oo~

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Chapter 4 -Integers (i.e. Signed Numbers)

4.1 Number Line and Conceits

Review the Basic Operations Adding and Subtracting sections 1.4 and 1.5

Number line: Graphical representation of the numbers. Mark the location of zero.

Mark the location of zero then plot the following numbers on the number line.

—2, 3, ~, and ~3

No homework for 4.1

4 2 Adding ~~~a ~y~}r~^'~~-~z Signed Numbers

Draw the number line. Mark the location of zero.

1. Start at the zero.

2. Go in the direction of the first number.

3. Travel in the direction of the second number.

4. The answer is the number where you ended up.

Example 1 3-~5

negative

Example 2 —3 + 7

Example 3 —8+5

—'plus

positive

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Example 4 5+(-9) plus times a minus is a minus

5 — 9

Example 5 (-2) + (-3) plus times a minus is a minus

Example 6 Use the number line to simplify the 6 expressions given below.

—12 + (-15)

8+9

17 + (-9)

(-24) + 17

Subtracting Signed Numbers

Subtracting is adding the opposite. OR

The opposite of a negative is a positive

The Opposite of a Negative is a positive.

5+(--12)

(-25) + (-33)

1. A plus tunes a minus is a minas.

(different signs produce a minus)

2. Every other multiplication is a plus.

i.e, a minus times a minus is a pCus

a plus times a plus is a plus

Example 7 Simplify the 6 expressions given below.

5-8 5+(-8) —4—(-3) —4+3 —5 — 6 —5 + (-6)

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Example 8 Simplify the 5 expressions given below.

4-12 -5-7 (-4)-(-7} 18-13 (~3)-5

Example 9 Simplify the 5 expressions given below.

-5-9 14-28 -13+8 16+24 -25-(-34)

4.2 Adding ~~„a ~~~'~}r^^*~r~' Signed Numbers Homework

For problems 1-6, show the addition on the number line.

1. 3 + 4 2. 5 + (-3) 3. (-2) + (-4)

Perform the indicated operation to simplify the following expressions.

7. S + (--14) 8. (-9) + (-8) 9. -15 + 6

10. 15 + 23 11. 15 + (-23) 12. (-15) + (-23)

13. 24-(-ZS) 14. -12-13 15. --12+13

16. (-8) - (-3} 17. l 4 - ~- 7~ 18. -- 23 + (-12)

19. 45 + (-34) 20. (-24} - (-22) 21. 15 + (-23)

22. 45 - 25 23. - 45 - 25 24. (-27) - (-22)

25. --52 - 25 26. -52 + 25 27. 52 - (-25)

28. (-24} + (-16) 29. (-24) - (-16) 30. 18 - (-9)

31. -24-36 32. -24-(--36) 33. 25+(-36)

34. 54-(-18) 35. -54+18 36. -76-85

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4.3 Multi~lyin~ and Dividin~Si~ned Numbers

Multiplying ~ repeated addition ~ 3 ~ 8 ~ 3 + 3 +...+ 3

8 times

Example ~ Multiply the numbers below.

2.55+510

5.22+2+2+2+210

4.7 ~ 7+7+7+7 ~ 28

6.3~

3.8~

Now considering signed numbers. Note: —(—a) = a

3•(-5)~-5-5-5~-15

4~(-2)~-2-2-2--2~—$

—3~4~

—5 • (-4) ~

—6~5~

Notice a pattern.

pos ~ pos ~ pos

Spring 20 ] 0

neg ~ neg ~ pos

OR if the signs are different ~ neg all other multiplication is positive

pos•neg ~ neg

neg ~ pos ~ neg

Multiplying signed numbers.

1. Multiply the numbers together.

2. If the signs are different ~ neg answer otherwise the answer is positive

Example 2 Multiply the signed numbers below.

5•(-8) (-3)•(-7) —S•4 12.5 15(-8) (-9)(3)

Dividing signed numbers is like multiplication.

If the signs are different ~ neg answer otherwise the answer is positive

a —=c~b•c=a

b

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Example 3 Divide the signed numbers below.

4 1 = 3 ~ 4.3=12 6 =~~

So, let us consider the following cases. What is the sign of the c -value?

- 

pos

= c ~ pos • c = pos ~ c =

pos

neg _ ~ ~ neg ~ c = neg ~ c

neg

pos

=cam neg•c=pos ~ c=

neg

neg _ 

c ~ pos • c = neg ~ c =

pos

Example 4 Divide the signed numbers below.

I8 -16 52 105 -24 -42

3 5

4 3 Mulfi'~lvin~and Dividin~,Signed Numbers Homework

Multiply or Divide as indicated.

1. S • 4 2. (-3) • 6 3. (-5) - (-3) 4. 4 ~ (--7)

5. 7(-5) 6. - 3 • S 7. 23 • (-24) 8. (-12)(-20)

9. (-15) ~ 4 10. (-8)(-7) 11. (-8) • 7 12. (-l4)(-15)

13. (-24) = 4 14. 3 S = (-7) 15. (--45) = (-5) 16. ~ 28 4

17. - 6~ 5 18. 9~ 3

19. 5~ 4 20. (-100) = 2S

21. (-75) :- (-25) 22. - 39 ~ 3 23. 144 :- (-9) 24. (-65) ~ 13

-85

25. 18 ~ (-5) 26.

5

27. (-65) ~ (-13) 28. 

-144

29. 25 • (-8) 30. (-35)(--4) 31, ~5~4

33. -36 ~ 4 34. (-13)(-15) 35. 13(-15)

-6

32. 5/--4

36. (-84~=(-6~

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4.4 Order of Operations

Powers (Exponents): 3$ = 3.3 • ... • 3 ~ repeated multiplication —,~—,

8 times

Example 1 Multiplication is repeated addition.

3.44+4-r4~12

5 - (-2) ~ (-2) + (-2) + (-2) + (-2) + (-2) ~ —10

4.6~

2 • (-7) ~

Example 2 Exponents indicate repeated multiplication

3~ ~3~3.3.3~81

(-5)3 ~ (-5) • (-5) • (-5) ~ —125

(-~)' ~

2io ~

Example 3 Simplify the following three expressions.

2+4.3~+5.7 4+32 +42 4•SZ -324

Order of Operations

1. Grouping: Parentheses () Brackets [ ] Braces { } numerator and denominator

2. Exponents (Powers)

3. Multiply ar Divide in order of occurrence from left to right

4. Addition er-~~tb~~ae-tom

Example 4

'~ 2 +4z

r~

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MAT 082 Basic Arithmetic Page 82 of 105 Spring 2010

4.4 Order of Operations Homework

Simplify the following expressions.

1. (4-6)z +(5-3)' 2. 5•(2-8)+3~(2+4) 3. 5+3.2'

4. (3-1)4 +(1+2)5 5'

z ~

115+2 6~

z a ( 1z ( 1z

6

7. 13 I +I S

225 l J ~ J

8.

~ 2)

3' S -4

9. 4.35 -3.2' 16.

3+2 15

(5 - 2)2 4 5 7

17. 5~(3-7)2 +2•(5-2)2

3

18. 5 +~ 20. (3-5)a •(2+3)z

21, 3 • ~ ~ ~ 4 I 23. -34.5 + (-5)z • 3 26. (-3)a + ~_S~z

J

27. -34 +(-S)2 28. (-3)a -Sz 29. (3.8-r 52)-(3z -4•S)

30.

g'3-5.6

31. -21-2+(-9) 32. --12--(-5)+(-3)

52 - 3z

33. -2(-6) + 7 34. -3 +4(6 - 9) 35. -20 ~ (-S)(-4)

36. -9Z -~--E 5)Z ~7. L35-63]

38. -3-2(4-6)

~~~4'2]

39. ~ ~1-2Z]~~ 40. -8+19-(-3) 4Z. -9-(-13)+(-12)-21

42. - g +01 43. -4(-9) + 2 44. -3 + 4(-2)

4S. -21.2(-14.5) 46. (-4)z 47. --52

213

48' ~

3J

MCCCD/Martinez00593

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~~~' ~iZD

G ~'' ~°

Ntal~atyua cav~dh~

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Chapter 5 -- Decimals &Percents &Fractions

S.1 The Basics about Decimals

° ~ b

:~ ~ ~

°

~ ~ ~ ~

v~ ~ v ~ b

~ ~~ O

-~

H

~v ~ ~ b b .~

N v ~ ~~ G

'G ~ ~ G ~ ~ ~

o

~ w

~

y

~

N

~ ~ ~ ~ ~ o

~ ~

x H ~ ~ xi -~ ~ H~ ~ x ~ E~ O~ vH

a~i

x ~ H x~

1 1 1 1 1 1

I 1 1 1 I

1 1 1 . 1

~ 1

1 1 1 1 1

1 1 -^~ I

Millions Thousands Ones These are fractions.

On the left side of the decimal, A decimal is written

each group of three digits has a name and between the whale

is separated by a comma. numbers and fractions.

~ decimal is either a proper fraction

.37 is the proper fraction 

00

or a decimal is a mixed number.

4.29 is the mixed number 4 0~ o

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The mode is the number thafi occurs most o en in a list.

Example 1: Find the mode far this list of numbers. 84, 90, 95, 98 and 88

Answer; The mode score for this list is

Example 2: The sales of photo albums at Juanita's Card Shop for each day last week

were $86, $149, $103, $118, $117, $126, $158 and $149. Find the mode.

Answer: The mode score is

Steps for Finding the Mode of a set of values (The value appearing the most times.)

1. Find the value that appears most often in the list of values. This is the mode.

2. If no value appears more than once, there is no mode.

3. If two different values appear the same number of times, the list is bimodal.

Example 3: Find the mode of the following three sets of values:

3 11 3 2 10 9 18 3 Answer:

3 17 3 72 2 10 9 1Q Answer:

3 1 11 12 2 10 9 15 7 Answer:

Summary:

The meatt is the average and is found by first adding all values to get the total then

dividing youx answer by the number of values. mean = 

add all numbers

number of values

The median is the middle number in a list of numbers when listed in order from

smallest to lamest.

The mode is the number that occurs most o en in a list.

1.12 Mean, MedianLMode Homework

Find the mean of the following data.

1. Tanya had test scores of 96, 98, 88, 82, and 92. Find the mean (or average) score.

2. These are several months of utility bills: $25.12, $42.58, $76.19, $32, $81.11, $26.41,

$19.76, $59.32, $71.18, and $21.03 Find the mean.

3. Find the mean for the following numbers: 74, 81, 39, 74, 82, 80, 100, 92, 74, 85.

4. What is the mean for these numbers: 51, 32, 49, 51, 49, 90, 49, 60, 17, 60?

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Find the median of the following data.

5. Tanya had test scores o£ 96, 98, 88, 82, and 92.

6. Follnwzng is a list of monthly utility bills: $25.12, $42.58, $76.19, $32, $81.11,

$26.41, $19.76, $59.32, $71.18, and $21.03

7. Find the median for the following numbers: 7~, 81, 39, 74, 82, 80, 100, 92, 74, 85.

S. What is the median for these numbers: 51, 32, 49, 51, 49, 90, 49, 60, 17, 60?

Find the mean and the median of the following lists.

9. This is a list of math scores: 84, 90, 95, 98 and 88

Z0. Here is this list of measurements: 178 ft, 261 ft, 126 ft, 189 ft, 121 ft, and 195 ft.

17. This is a list of prices: $7, $23, $15, $6, $18, $12, $24

12. Here is a list of ages, in years: 74, 7, 15, 13, 25, 28, 47, 59, 32, artd 68.

Find the mode for the following list of values.

13. Tanya had test scores of 96, 98, 88, 82, and 92.

14. Monthly utility bills of: $25.12, $42.58, $76.19, $32, $81.11, $26.41, $19.76, $59,32,

$71.18, and $21.03

15. Find the mode for the following numbers. 7~ 81, 39, 74, 82, S0, 100, 92, 74, 85.

16. What is the mode for these numbers: 51, 32, 49, 51, 49, 90, 49, 60, 17, 60?

17. This is a list of math scores: 84, 90, 95, 98 and 88

18. Here is this list of measurements: 178 ft, Z61 ft, 126 ft, 189 ft, 721 ft, and 195 ft.

19. This is a list of prices: $7, $23, $15, $6, $18, $12, $24

20. ~arnily member's ages, in years: 74, 7, 15, 13, 25, 28, 47, 59, 32, and 68.

Find the mean, the median, a.nd the mode of the following lists.

21. A List of the age of part-time employees (in years): 28, 16, 22, 28, 34, 22, 28.

22. Total points on a health screening exam: 312, 219, 782, 312, 219, 426, 507, 600.

23. Monthly commissions of salespeople: $1706, $1284, $1653, $1892, $1301, $1782.

24. A set of numbers: 482, 485, 483, 485, 487, X87, 489, 486.

25. Monthly salaxy of a college president. $10,708; $11,519; $10,972; $12,546; $13,905;

$12,182.

26. These are Maria Montano's test scores in her arithmetic math course:

93 76 83 93 78 82 87 85

from arithmetic book fall ?009

Find the mode.

1 I , 26, 34, 43, 26, 51 12. 17, 7, 11, 11, 14, 17, 18

13. 0.2, 0.2, 1.7, 1.9, 2.4, 0.2 . 14. 700, 700, 8Q0, 2700, 800

15. $14, $17, $21, $29, $17, $2 16. 20, 20, 20, 20, 20, 500

17. One summer, a student earned the following amounts over afour-week period: $102, $112,

$130, and $98. What was the average amount earned per week? The median?

18. Gas Mileage. A 2001 Ford Focus gets 528 mi of highway driving on 16 gal of gasoline.

What is the gas mileage?

19. To get an A in math, Marcus must score an average of 90 on four tests. His scores on the

first on the first three tests were 94, 78, and 92. What is the lowest score that he can make on the

MCCCD/Martinez00597

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~ w o ass~v~

~cc~ ~s ~v~ ~s ~ s

~,ti ~. c ~ ~,U a v ~ c, v~ ~

a s ~ Iry o ~ s

~ ~ ~ ~~ a -~~ ~

~ Iry n a cc~ ~s ~v~

vac. ~ n u. ~~ ~ ~st~~

a ~ ~ws t~ ~ ~s ~~a

> >

~~o ~~a ~~~, w~~ ~,~.

Mart~w Lu.tl~er l~C~w~,Jr.

MCCCD/Martinez00598

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Chapter 2 -- Fractions

2.1 Reducing (Sim~Ii ing) Fractions

Drawing fractions

Fraction ~ parts of a wholes ~ ~ 3 ~ 2 parts out of 3 total

2

Example 1: Draw ~ of the whole circle.

2

— ~

3

Example 2:

Show 4 parts out of a total of 6 below.

a total of 3.

4

6

Reducing Fractions

8

12

Example 3.

Show 8 parts out of a total of 12

2 1

6 3

G —

8

2

— ~

3

1

2

Example 4:

Show 2 parts out of

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Factors are numbers which are multiplied. For example: the factors of 6 are 2 times 3

We reduce fractions by reducing common factors between the numerator and denominator.

This means that fractions like ~ or 3 are reduced to 1. What other fractions can be written as 1 ?

To reduce 2/6, 2 is written as the factors 2.1 To reduce 6/9, 6 is written as the factors 2.3

and the 6 is written as the factors 2~3 and the 9 is written as the factors 3.3

2 2.1 1 6 3.2 2

6 2~3 3 9 3.3 3

Example 5: Reduce these fractions by first writing in factored form then reduce to ones.

14 ~ 15 ~

35 39

56 ~ 52 ~

Reducing nsrng primes

Prime numbers are numbers where the only factors are one and itself.

What are some of the prime numbers?

Example 6: Use primes to reduce fractions.

Breaking numbers to prime form or

36

42

using one prime at a time.

36

42

MCCCD/Martinez00600

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Example 7: Use primes to reduce fractions.

24

90

Example 8: Use primes to reduce fractions.

45

iaa

2.1 Reducing Fractions Homework Use primes to reduce fractions.

1 g 2 1S 3 45 4 105

12 25 75 280

5 54 6 210 ~ 168 g

90 510 2l6

9. 182 10. 240 11. 12.

390 400

Rewrite the fractions so they have the indicated denominator.

13. ? ~— 14. ~ ~ — 15. 3 ~

5 IS 3 27 7 42

What is the fraction represented by the following?

17. .5 18. .1 19. .25 20

21. .75 22. .10 23. 50% 24

25. .50 26. 25%

270

315

I6. 

9 ~ 63

.3

75%

MCCCD/Martinez00601

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2.2 Adding and Subtracting, Fractions

To add or subtract fractions we must have a common denominator.

u

1 + 2 ~ 3

4 4 4

Formula for adding fractions with common denominators

a ~ b ~ a + b Add the numerators and carry the common denominator.

c c c

Example 1; g + g ~

Example 2: 5 + 3 ~

12 12

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What if we do not have a common denominator?

1

4

1

3

3 + 4 ~ 7

12 12 12

Formula for adding fractions with different denominatflrs

(Assuming there are no common factors in the denominator)

a c a d c b a• d+ b~ c other denominator

— -F- — ~ — • — + — • — ~ multiply by the

b d b d d b b• d other denominator

otherwise you need to multiply by the 

lowest common denominator

lowest common denominator

Example 3 Add the fractions

(no common factors in denominators)

2 1

—+-

5 4

Example 4 Add the fractions

(denominators with common factors)

3 7

— + ---

8 ]2

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MAT 082 Basic Arithmetic Page 38 of l05 Spring 2010

Subtraction is exactly like addition, We will also need a common denominator to subtract.

Formula for subtracting fractions with common denominators

a b a—b

— — — ~

c c c

Formula for subtracting fractions with different denominators

a c a•d—b•c _. _ ~ ~

b d b•d

Simplify:

Example 5

11 3

12 12

Example 6

5 3

6 8

2.2 Adding and Subtracting Fractions Homework

Example 7

7 4

ld W 1S

Preliminary work. Write the prime factors for the following numbers.

a. 8 b. 12 c. 6 d. 1 S e. 14

h. 18 i. 52 j. 16 k. 32 1, 45

o. 19 p. 35 q, 40 r. 115 s. 51

v. 77 w. 28 x. 23 y. 49

Add or subtract the fractions as indicated.

1. 3+1 2. 5+1 3. S+?

4 3 8 6 12 15

5. 3+z 6. 4-1 7. 3—~

10 9 5 6 8 10

5—~ ~ 4

13 + 5

9. 10. — 11.

7 9 18 21 21 18

13. 12-5 14. g-3 15. 5+~

17 ]9 15 20 24 20

17. 5—3 18. g+8 l9. 5+~

16 12 35 21 12 16

21. 5+~

16 20

4.

8.

12.

16.

20.

f. 36

m. 78

t, 92

2+ 3

5 7

9_3

13 14

8 y 4

IS 25

23 17

—+-

52 78

14 T 6

2S 35

g. 2

n. 20

u. 56

MCCCDlMartinez00604

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MAT 082 Basic Arithmetic Page 39 of 105 Spring 2010

2 3 Multi~lvin~and Diviclin~ Fractions

Fraction ~ a ~ a = b Fractions can be read like the division of two numbers.

b

Multiplying Fractions (Assuming there are no common factors in the denominator)

Multiply the numerators and multiply the denominators.

a c _ a ~ c Otherwise,

b d r b•d

1. Factor the numerator and the denominator

2. Reduce common factors to one

3. Multiply the numerators and multiply the denominators.

Simplify:

Example 1: Example 2:

3 2 5 3

5 6 6 g

Example 3:

12 25 ~r 12 25

35 42 35 42

Example 4: Example 5:

4 9 12 26

21 20 39 63

MCCCD/Martinez00605

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MAT 082 Basic Arithmetic Page 40 of 105

Dividing Fractions ~ Multiplying by the reciprocal

a _ C ~r ~_a d

b d ~ b c

~' T T

copy change flip

a d 

Now do the multiplication.

b ~ c

Example 6: Simplify Example 7: Simplify

S 3 14 24

6 ~ 8 15 35

k'irst: copy-change-invert

Second: factor the numerator

and the denominator then

reduce to one where possible.

Third, multiply.

Example 8:

12 25

35^42

Spring ZO 10

2.3 Multiplying and Dividing Fractions Homework

Perform the indicated operations on the fractions

1. 3. 5 Z. 4. 6 3. 12 21 4 13 10 S 14 5

4 6 9 20 3S 24 25 39 23 21

6. 3 • 11 7. 6 •~ 1 8. 3•~ 9. 

20 35 

10. 

34 2l

22 IS 55 33 5 7 21 32 49 51

22 25 16,12

11. 12. 13. ~-4 14. 5-3 15. g-3

45 33 23 35 3 7 8 14 15 25

24 8 26 39 12 9 46 115 34 Sl

16. --=— 17. -- 18. --- 19. —

~

, 20. --

35 21 4S 40 25 35 5Q 24 48 92

42 34 15 12 14 3S 5 15

21. —=— 22. - 23. - 24. ---

55 77 28 35 25 56 18 28

MCCCD/Martinez00606

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MAT 082 Basic Arithmetic Page 41 of I OS Spring 2010

2.4 M~txed Numbers mixed number ~ improper fraction

numerator

Fraction ~

denominator

Proper fraction ~ numerator is smaller than the denominator 3

Improper fraction ~ numerator larger or same size as denominator 3 or 3

Mixed number ~ whole number together with a fraction 5 7

Drawing Example 1 Split each whole circle into 4 equal pieces and shade 13 pieces.

Each piece is ~ . There are 13 pieces shaded ~ ~ .

There are 3 wholes shaded and one-fourth of a whole ~ 3 4 .

Example 2 Draw five halves and write this two ways: as a fraction and a mixed number.

Example 3 Draw 3 5 and white as a fraction.

3

Converting from a mixed number to an improper fraction. ~ g

1. Multiply the whole number by the denozxtinator. —~ 4(8)

2. Add the numerator; this is the new numerator} -~ 32-x-3

new numerator 35

3. Write as an improper fraction. —> —

denominator 8

MCCCD/Martinez00607

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MAT 082 Basic Arithmetic Page 42 of 105 Spring 2010

Exannple 4 Write each mixed number as an unproper fraction.

53 35 2~

Steps for converting (chan~in )~ from an improper fraction to a mixed number.

33

1. Divide the numerator by the denominator. (Long Division) 6

quotient

denominator numerator 6 33

2. Write as a fraction.

remainder

quotient

denominator

Example 5 Write each zmproper fraction as a mixed number,

17 34

5 7

63

5

3 1

5 - or S -

6 2

234

11

2.4 Mired Numbers Homework

Convert to mixed number form.

1. 8 2. 9 3. 61 4.

35

5.

25

3 4 5 8 7

75 14

6. 7.

125

8. 9.

39 10.

335

lZ 5 7 9

17

Convert to improper fractions

11. 35 12. 5~ 13. 29 14. 9~ I5. 5~

16. 125 17. 3 ~ 18. 12 5 19. 93 20. 23S

6 10 11 8 9

Reduce and convert the fractions

21. 25 22. 

270

23. ~5 24. 90 25.

24

15 126 12 75 16

26. 

22S 

27. ~5 28. 84 29.

150

30.

234

100 21 72 36 52

MCCCD/Martinez00608

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2.5 Operations with Mixed Numbers

The four main operations are

Steps for Adding Mixed Numbers or subtracting mixed numbers

1. Convert mixed numbers to improper fractions. or 1. Add the whole numbers

2. Perform the operation. 2. Add the fractions

3. Convert back to a mixed number, if necessary. 3. Write the answer as a mixed number.

Example 6: Simplify

23+41 ~ 13+25 234 1

5 6 5 6 5 6

~ 13.6+5.25 ~ 78+125 ~ 203 2~4 ~ 3 + 1

5~6 30 30 5 6

6 3~6 1.5

6 + — + —

30 203 5.6 6~5

—180 ~ 6 23 

6 } 23 ~ 6 23

30 30 30

23

Example 7: Simplify Example 8: Simplify

6~ —21 95 —34

1S 6 12 15

6+? — 2 — 1

15 6

4 + ? — 1

]5 6

2 6 1 5

4 +

15 6 6 5

12 5

4 +

_

30 30

4 + 7

30

4~

30

3.5 2.3

LCD is 3 ~ 5.2 = 30

18+5=23

MCCCD/Martinez00609

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Example 9 Simplify the expressions

1_ ~ 3— 3 8-21 8+21

5 4 4 4

1Viultiplvin~ or Dividing Mixed Numbers Note: First rewrite the mixed numbers as fractions.

Example 10 Example 11

23 •41 23 =41

5 6 S 6

Example 12 Combine the mixed numbers by performing the indicated operations.

65-3 ~ 75-43

8 12 6 8

2.5 Operations with Mixed Numbers Homework

Perform the operations indicated ,

1. 31 +21

4 3

2. 5~+13

3 5

3. 6?

5

-31

4

5. 35+23 6. 21 •3? 7. 5 1 3~

6 8 4 5 4 7

9. 3l :-2~ 10. 12~=84 11. 25=35

7 5 3 9 6 8

13. I 13=117

15 25

14. 7~ :-3 5

7 21

15. 61'+55

4 6

17. 2 S —121 18. 7 1 •5 ~ 19. 3 

4 317

14 34 22 15 15 21

4. 2~-3

7 4

8: 7 1 4 2

6 5

12. 84=2~

S IS

16. 3 ~—1 9

12 16

20. 6 9 = 8 4

14 7

MCCCD/Martinez00610

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MAT 082 Basic Arithmetic Page 45 of I05 Spring 2010

2.6 Form A: Addition and Subtraction

1.31+83+53 2. g-1+5 3.21-83-~-5 4. 8+13 3 -1

2 4 4 3 9 7 3 4 27 9

5. ~-83+5 6. $+13~-1 7. 5+~-24 8. 6~-3~+9?

3 4 2 9 9 5 S 5 5 5

9.11+3 10 24-3 4 11. 11+3 12. 

5+8`14

6 8 11 6 4 9 3 5

32 2 47 1 5 2 8 2 1 5

13. ----3-+— 14. 8-+3- 15. 25-25-+- 16. -+---

5 5 5 4 6 3 7 3 19 2

3 18 1 3 3 2 1 5 1 5

17. 5-9+ ~ 18.32+84-+-5~ 19. 2~--9+b 20.84+36

21. ~+1-21 22. 3-9+5

3 9 2 5 7

MCCCD/Martinez00611

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MAT 082 Basic Arithmetic Page 46 of105 Spring 2010

2.7 Form B: Multiplication and Division

4 4 I7 88 3 2 15 225

9 9 3 121 7 63 42 18

7 1

5 4—=2—

4

6 S—•—

33 1 1

7.11—~5—

80 22

8. —=-

11 2 9 2 2 4 140 133

9.61 -3 1Q. 23•l~ 11. 4=15 12. —

5 49

2 4 5 7 7 8 7 65

3 3 80 135 1 2 1 —2

13. 13—•8— 14. —: — 15. 5—•2— 16. —2--:--

5 2 140 22 2 5 3 3

17. ~ • S 18. —3=--6 19.54.84 20. 

7_ 88

35 24 8 32 9 9 3 121

2I. 

31 12 22. ~ =-2 23. 4 ~ 5 24. —

3 ~-6

2 5 —3 3 35 24 8 32

MCCCD/Martinez00612

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MAT 082 Basic Arithmetic Page 47 of 105 Spring 2010

2.8 Form C: Mixed Problems

7 4 S 17 8 1 1 8 S 15

1. 3+129• 2. 2 =3+18 3. 5+73 9 4. 3+3

6~ 4

7 7 1 1 Z 2 1 2 1 1 19-4 5

5. ---4—=2—+5— 6. 6— : 2--4— 7. 5----6---.— 8. =1-

13 6 2 4 2 3 3 5 9 S 3+7 8

1-9 57 3 2 4 27 9 39 2 1 1

9. •— 10. $—=2--7— 11. --6—•— 12. —+1—•6-

7+5 3 2 3 7 5 8 2 3 2 3

1 1 8 3 19 1 9 2 1 1

13. 2--: —~-3 14. —+—•— 15. —+1—=I4 16. —+2—~-

3 4 15 Z 3 6 2 5 2 7

2 8 8 3 19 16 4 1$ 1 3 1

17. —~—+18 18. —+—•— 19. —~—+-- 20. —+—•6-

51 3 15 2 3 3 33 3 3 2 3

21. ~ +5 1 -7

6 2

MCCCD/Martinez00613

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 220 of 243
Case 2:12-cv-00702-DGC Document 68-7 Filed 10/11/13 Page 88 of 187

NtAT 082 Basic Arithmetic Page 48 of 105 Spring 2U ] 0

2.9 Ratios

A ratio compares one number with another. Write ratios as fractions reduced to lowest terms.

Ratios can be written in several different ways. If we compare the number 6 with the number 7,

the ratio is read as "6 to 7" and is written 6 or 6:7 ar 6 to 7. We are going to use

7

the fraction style.

If two numbers with units are being compared (like 7 inches to 1 foot), it is important to state

the units of the numerator and the denominator clearly: ~ 

inches

1 foot

7 inches

In fact, if passible, make the units the same; You are comparing the same unit.

12 inches

Remember that ratios remain fractions but must be reduced to lowest terms.

Example 1: Compare the number 20 with the number 37.

Answer: The ratio is

Example 2: Compare 1 dollar to 5 quarters.

Answer; ~e ratio is

Example 3: What is the ratio of 12 inches to 1 S inches?

Answer: The ratio is

Example 4: What is the ratio of 5 to g ?

MCCCD/Martinez00614

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MAT 082 Basic Arithmetic Page Q9 of ] OS Spring 2010

2.9 Ratios Homework

Give the ratio of the following quantities.

1. 8to6 2. 6to8 3. 64 to 12 4. 12 to 64

5. 100 to 250 6. 250 to 100 7. 13 to 26 8. 36 to 18

9. 3tol 10. sto3 11. ~to6 12. 9to

11

4 4 8 8 3 3 5 5

13. 6 to 6 14. 5 to 1 15. 2 1 to3 1 16. 5~ to13 4

5 7 3 3 2 2

17. 2 2 3to 3 18. ~ to3 ~ 19. 0.05 to 0. Z 5 20. 0.21 to 0.03

21. 0.3 to 3 22. 0. 5 to 10 23. 1.2 to 10 24. 6.4 to 0.8

25. ~ to 1.5 26. 1 to 0.75

2 4

27. Regarding the diagram below, AC represents the length of the line segment that starts at A

and ends at C.

From the diagram we see that AC = 8.

U

A 8

a. Find the ratio of BC to AC

b. What is the length AB?

~ c. Find the ratio of DE to AE.

28. A family of four budgeted $400 for food, $100 for gas, $150 for utilities, and $250 for the

house payment.

a. What's the ratio of the house payment to the food bill?

b. What is the ratio of the gas bill to the food bill?

c. What's the ratio of the utilities bills to the food bill?

d. What is the ratio of the house payment to the utilities bills?

MCCCD/Martinez00615

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MAT 082 Basic Arithmetic Page 85 of 105 Spring 2010

A decimal written in fraction form has a denominator with a power often.

~

o

°'

'~

._

.~ ~

~

~

ro ~

~ ~

.a

~ ~

,V'

v

p

'" ~'

'b ~ ~ `C7

a~ ~ o ~

'ON

v

'b

a

C

ro

~ 'i3 '~ ,~

o ~; G ~

b ~ o

~

b `~° ~ ~ ~v ~, ,~ b

~

~

°

~ Ts `~° ,o

~ ~ o

x

~

F ~

~ o v °

x~ E-~ ~

~

x

~

E--~

~

O

~

H x ~

v

E-~ x~ ~

1 1 1 1 1 1 7 1 1 . 1 1 1 1 1 1

Millions Thousands Ones T~tESE aTe fTdChOriS.

1 1 1 1 1 1 1 1 1

10 lOZ ~r 100 103 or 1,000 104 

or 

10,000 105 

or 

100,000

ten- hundredtenths hundredths thousandths thousandths thousandths

The exponent tells how many zeros are in the number, In 14 ~ the exponent is 2 and the 100 has 2 zeros.

tenths hundredths thousandths ten-thousandths hundred-thousandths

The location of the number (based on the decimal point} gives the value of the number.

The denominator is a po~rrer of 10 i.e. 10, 10 2 , 10 3 , 10 ~ , 10 5 etc.

For example: the 5 goes under the

10 in the number five-tenths • 5

The 7 goes under the 100 in the • ~ 7

number seven-hundredths

MCCCD/Martinez00616

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MAT 082 Basic Arithmetic Page 86 of 105 Spring 2010

Example: Write the number

forty-three thousandths •

Write the number twenty-five hundredths.

Write the number two hundred forty-three and eighteen ten-thousandths.

Write the number one thousand thirty-six and ninety-fzve thousandths.

Write the number 37 and 50 hundredths.

Read the number ~. 4 5 7 • ~ 6 3

Answer: one thousand four hundred fifty-seven and sixty-three thousandths.

5.1 The Basics about Decimals Homework

WFite each fraction as a decimal or a mixed number.

7 9 27 39 2 4 2 4 29

1.— 2.— 3.— 4. — S.— 6.— 7. 5— 8.6---- 9.-

10 10 100 100 5 5 S 5 S00

3 1 1 401 203 601 1003

10. — 11, 7- 12. 9 13. 15 14. 11— l5. 16.

125 125 200 500 500 2000 5000

17) 6+ 4+ 3 + g 1 S) 3+ ~+ 8 + 4 19) 9+ 3

10 100 1,000 10,000 10 100 1,000 10,000 100 1,000

3 8 4 6 0 3 6 8

20) 68+—++ + 21) 491+—+---+ +

10 100 1,000 10,000 10 100 1,000 10,000

22) 436 + ~ + ~ + ~ + ~ 23) 

483 24) 8 + 2

10 100 1, 000 10, 000 10, 000 1, 000 10

MCCCD/Martinez00617

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MAT 082 Basic Arithmetic Pale 87 of 105 Spring 2010

Write each number in words.

25. 2.8 26. 5.1 27. 9.08 28. 12.06 29. -705.625 30. -804.399 31. O.Q046

32. 0,0083 33. 5.62 34. 9.57 35. 16.23 36. 47.65 37. -.205 38. -.495

39. 167.009 40. 233.056 41. 3000.04 42. 5000.02 43. 105.6 44. 410.3

Write the words as numbers

45. four hundred ten and three tenths

46. Two and eight tenths

47. Five and one tenth

48. Nine and eight hundredths

49. Twelve and six hundredths

50. Negative seven hundred five and six hundred twenty-five thousandths

51. Negative eight hundred four and three hundzed ninety-nine thousandths.

52. Forty-six ten-thousandths.

53. Eighty-three ten thousandths

54. Five and sixty-two hundredths

S5, nine and fifty seven hundredths

56. sixteen and twenty-three hundredths

57. forty-seven and sixty five hundredths

58. Negative two hundred five thousandths

59. negative four hundred ninety-five thousatadths

60. one hundred sixty-seven and nine thousandths

61. two hundred thirty-three and fifty-six thousandths

62. three thousand and four hundredths

63. five thousand and two hundredths

64. one hundred five and six tenths

65. Read the following numbers and then write them in exponent form where 10 is the base:

10 lOQ 1,000 10,000 100,000 1,000,Q00

66. Multiply the following and see if you can see ar~y pattern.

3.S(100) .04(10) 27.396(1000) 142,789(100) 5.0432(10)

Read the following numbers and write them as fractions or mixed numbers.

67) .6438 68) .3084 69) .Q93 70) 68.3846

71) 491.0368 72) G36.0000 73) .0483 74) .208

MCCCD/Martinez00618

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MAT 082 Basic Arithmetic Page 88 of ]OS Spring 2010

5.2 Chan~in~ Decimals, Fractions, and Percents

Changing from decimal to fraction is easy since all you do is write the number as it is

read, then reduce the fraction where possible.

Example .5 is read asfive-tenths.

5 1

So it is written as the fraction ~p and then reduced to 2

Chan~in~ from decimal to percent is also easy since percent means per 100.

This means that in percent the denominator is always 100.

50% is 50 or 1 1 QO% ~S 100 

or 1 100% is the same as the number I .

100 2 100

25

Example 1 .25 is read as twenty-five hundredths. So it is written as the fraction ~ 00

This is 25 per 140; that is 25 percent which is 25%.

.25 (100%) = 25%

Example 2 Change ,72 to a percent.

Answer: ,72 is read as So it is written as the fraction

X115 15 W~11C~1 15

Example 3 Change .385 to a percent.

.385 times 100% (.385 times 100)%

MCCCD/Martinez00619

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 226 of 243
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Mt1"I' 082 Basic Arithmetic Page 89 of 105 Spring 2010

Chan~in~ from percent to fraction

Chan~in~ from percent to decimal

Chan~in~ from fraction to decimal

Chan~in~ from raction to percent

MCCCD/Martinez00620

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MAT 082 Basic Arithmetic Page 90 of 105 Spring 20 ] 0

Decimal Fraction Percentaee

1 = 1. = 100%

3 = .33 3 = 33 3 °/a

4 = .25 = 25%

5 =.2 =20%

g = .125 =12.5%

5.2 Changing Decimals, Factions, and Percents Homework

Write the following decimals as fractions.

1. 28 2. .3171 3. 97.08 4. .081

S. 4.365 6. 36.005 7. 4.3 8. 14.3684

Write each percent as a decimal

9. 65% 10. 125% 11. 1.2%

15. 22% 16. 44% 17. 530%

2l. 28% 22. 11.15% 23. 300%

Write each decimal as a percent\

27. 0.41 28. 0.62 29. 0.06

33. 0.736 34. 0.457 35. 0.028

39. 3.00 or 3 40. 5.00 or 5

Write each fraction as a percent

12. 60% 13. 0.6% 14. 0,8%

1 S. 270% Z 9. 5.6% 20. 1.9%

24. 900% 25. 70% 26. 80%

30. 0.03 31. 1.00 or 1 32. 1.36

36. 0.014 37. 0.006 38. 0.009

41. 0.3258 42. 0.7218

43. ~ 44. ll 45. 1 46. 1

25 50 2S 50

47. 9 4$. 3 49. ~ or 13

200 40 4 4

11 3 1 7 31 31 179 7

50.-or2- S1.- 52. 53. 54. 55.-- 56.-

4 4 16 SO 300 400 800 32

MCCCD/Martinez00621

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 228 of 243
Case 2:12-cv-00702-DGC Document 68-7 Filed 10/11/13 Page 96 of 187

MAT 082 Basic Arithmetic Page 91 of 105 Spring 20 ] 0

5.3 Adding &Subtracting Decimals

When Adding (or Subtracting) Decimals, Line up the decimal point and the numbers

according to their place value.

5.3 Adding &Subtracting Decimals Homework

Add the following decimal numbers.

1. 2.1 + 4.0 + 5.6 + 7.7

2. 64.38+7.76+3.I+629.34

3. 154.58 + 22.24 + 145.06 + 1.98 + 28.83

4. 234.12+86.3+2.71+81.67+95033+822+9.76

5. 1.0719 + 17,0034 + 69.901 + 6,3027 + 45.4545 + 4.9669

6. 35,6825 + 4.3678 -i-15.6293 + 23.8471 + 3.6325 + 49.1873

7. 651.07 + .4592 + .893 + 259.24 + 6.2432

8. 7.534 + 77.234 + 362.28 + 1.2457 + 32.567

9. 138.62 + .46235 + 1.4928 + 16.368 + 29832

l0: 3.721+4.93+187.45+9367+.673+1.0079+14.7

11, 779.642 + 9.67 + 43.984 + .3468 + I2.56 + 687.29 + 30.09

12. 698.37 -+- 2.9482 + .32164 + 2984.5 + 84.375

13. 543.7 + 98.07645 + 2345 + 34 + .2 + 16.9 + 87.87

14. 2413.35 + $096 + 2 + .0000865 + 12.Q06 + 90.09

15. .000000007 + 3 + 423.65 + 900.0006

16, 132435.46 + 6342.35124 + 7452 + 978.142 + 13

17. 24.3+35.6+.008612+1+1.89+6.80+73

18. 14423.3 + 5598.7098 + 25 + .0008 + 3.0001 + .O1

19. 235.5555 + 34.444 + 3.23 + 4.1 + 5 + 69875

20. 55,698 + 2 + 24536.1 + 985674 + 32.6 + 12

21. 3542 + 6856 + 37764 + .000008 + .23

22. 33.32 + .9900001 + 76$.49 + 3

23 78.99999 + .2 + 365243 + 32 + .00002 + 78.1

24.54.98 + .Ol + 57 + 3 87967 -i-100000032 + .3

Subtracting Decimals

Subtract the following decimal numbers.

1. 13.5836 8.7397 2. 252.3 18

5. 35.007 -13.06 6. 200.4 87.895

8. 823,106 507.836 9. $9.9 24.3

12. 39.65 23.95 13. 283.78 36.89

16. 647,1 347.869 17. 16.4 -12.625

20. .0098 .000Ob 2Z. 534.0009 2.09

23. 4.1 .OQ978 24. 6 --1.423

3. 600 - 507.36 4. 435.6 - 21.54

7. 5194.362 - 486.089

10. 94.5-47.8 ll. 45-14.78

14. 36 - 24.91 1 S. 436.275 - 9.839

18. 12.9- 3.1 19. 4-.0001

22. 65.89 - 43.0008

MCCCD/Martinez00622

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MAT 082 Basic Arithmetic Page 92 of t OS Spring 2010

5.4 Mulfiplyin~ Decimals

Multiplying Decimals Multiplying decimal numbers and multiplying whole numbers is the

same except for the answer's decimal point placement. Count the number of digits to the right of

the decimal in the first factor and in the second factor. The sum of these digits is the number of

digits to the right of the decimal point in the answer.

Place Value

Fraction denominators and decimal denominators

Example Multiply (3.432)(1.2) There are three digits in the first factor and one digit in the

second factor. This means there must be four (4) digits to the right of the decimal in the answer.

(3.432)(1.2) is 4.1184

Example Multiply (.1206)(.0034}

S.4 Multiplying Decimals Homework

Multiply the following decimal numbers.

1.23x2.51 2.2.3x.251

4.0.234x5.5 5.36.2x0.76

7. 3 x .835 8. .2 x 234

10. 42.8 x 200 11. 24.6 x 1.004 x 2.007

13. 65.009 x 2.41 14. 834.1 x 23 x 87.9

16. 543.21 x 54 17. 3.4 x .009 x 12.5

19. 54.3x234x2.3 20. 87x88x.1

22. 64.7 x .0008 x .08 23. 55.3 x 2345

3.

6.

9.

12

15

18

21

24

answer is .00041 Q04

.003 x 2b.32

3.926 x .128

5.24 x 1.97

32x0.12x43.1

1234 x .1234

76.2x43x2.I

534.5x22x.312

624.46 x .21 x .008

MCCCD/Martinez00623

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MAT 082 Basic Arithmetic Page 93 of 105 Spring ZOlO

5.5 Dividin~Decimals

Dividing Decimals The divisor must be a whole ntunber. If zt is not, then move the decimal in

the divisor to the right and move the decimal in the dividend the same number of places to the

right. Next place the decimal point for the answer directly above the newly located decimal

point in the dividend, Now divide using the steps for long division.

Example: Divide .21 4.321 Answer: 20.57

5.5 Dividing 17ecimals Homework

Divide the following decimals and carry your answer at least two decimal places.

1. 6.82 = .21 2. .8402 = 6.2 3. 824 = .21 4. 326.2 = 20.4

5. 93.66 = .602 6. 828.3 ~ 6.42 7. 7.832 -: 42.6 8. 1.72 = 21

9. 9.865 = 3.48 10. 106.82 = 3.971 11. 98.6 -:.83 12. 483 ~ 2.G

13. 876.987 = 4.32 14. .0072 ~ 2.09 15. .0009 = 33 16. 5425.6895 :- 643.54

17. 46.24 = 35.2 18. 867.0083 = 65.781 19. 764.98 = 23.65 20. 425.9978 = .0054

MCCCD/Martinez00624

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MAT 082 Basic Arithmetic Page 94 of ]OS Spring 2010

5.6 Percent Problems

1. What is 32% of 84?

2. What percent of 90 is 18?

3. 56 is 70% of what number?

4. What percent is 13 of 85? (Another way to ask this would be: What percent of 85 is

13?)

5. What is 16.2% of 85?

6. A computer system is reduced $495 froze the original price. What is the percent decrease?

7. What number is 62% of 1200?

MCCCD/Martinez00625

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MAT 082 Basic Arithmetic Page 95 of 105 Spring 2010

8. 425 is 85% of what number?

9. 38 is what percent of 5700?

10. 1950 is what percent of 5000?

11. 210 is 250% of what number?

12. What number is 33 3 % of 516?

13. 24% of 300 is what?

15.. .9% of 2000 is what?

17. 25% of is 10.

14. Find 320% of 60.

16. 30 is % of 50.

18. is 20% of 50.

MCCCD/Martinez00626

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MAT 082 Basic Arithmetic Page 96 of 105 Spring 2010

5.7 Miscellaneous Signed Numbers &Critical Thinking .Homework

Please answer with clear sentences.

1. What is the total purchase price if 2S acres are purchased for $380 per acre?

2. How much is missing from the PTA's fund if $832 remains from the original $1,405?

3. How much did the burglar get if he took three envelopes containing $129, $3,$27 and $45,

zespectively?

4. How much can the investor expect to get back if she is promised a 1,OQ0 percent return on a

$40 investment?

S. The average APS customer can expect her new monthly bill to total how much if the current

average bill of $110 is increasing by 8 percent?

6. What percent of the rancher's property has been sold if he has 200 acres remaining from the

original 1,000 acre property?

Sinnplify the following expressions.

7. 4-9 8. 4-~ 1 9. --~+3 1Q. -12-(-3)--9 11. 

--6--1-4J

12. 100 - 98 13. (-2) + (-8) 14. (-4) + 7 15. (14) + (-2) 16. 8 + (-6)

17. (-7) + 7 18. -20 -10 19. 5 - 8 20. -3 - 2 21. 

440

(-z)(-5)

22. 3 - (--4) 23. -4 -1 24. 10.0 25. 4 • (-5) 26. (-3) • (--5)

27. (-1)4 28. (--2)(-1)(-3) 29. -10 30. 6 31, -1-(-9) 32. -4

-5 -2 12

MCCCD/Martinez00627

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Case 2:12-cv-00702-DGC Document 68-7 Filed 10/11/13 Page 102 of 187

MAT 082 Basic Arithmetic Page 97 of 105 Spring 2010

e are vt~ a

~v~e re ea e

o.

xce en.c~

eze are

is xlo an ac u

a a ~.

Aristotle

MCCCD/Martinez00628

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Case 2:12-cv-00702-DGC Document 68-7 Filed 10/11/13 Page 103 of 187

MAT 082 Basic Arithmetic Page 98 of l OS Spring 2010

Chapter 6 —Unit Conversions &Graphs

6.1 Unit Conversions Length----Area—Volume---Liquid—Weight Temperature

Measurrn~ Length English System - U.S. Metric System -Other Countries

(Units are based on powers of ten)

Dimensional Analysis A procedure using unit fractions to convert from one unit of measurement to another.

Unit Fractions- Fractions where the numerator &denominator contain different units &the value

24 hours

of the uzut fraction is 1. Example: 

1 d aY ~

USA System

12 inches. = 1 foot

36 inches = 3 feet = 1 yard

5,280 ft. = 1 mile

Example Convert 2 months to seconds.

60 seconds = 1 minute

60 minutes = 1 hour

24 hours = 1 day

36S days = 1 year

12 months — 1 year

Convert 2 years to minutes.

6.1 Conversions Homework: Use dimensional analysis to convert the quantity to the indicated

unit rounded to two decimal places.

1) 100 in. to ft. 2) 100 ft. to in. 3) 21 in. to yd. 4) 2I yd. to in.

5) 12 yd. to ft. 6.) 12 ft to yd 7) 19,800 ft to mi 8) 0.25 mi to ft.

MCCCD/Martinez00629

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Case 2:12-cv-00702-DGC Document 68-7 Filed 10/11/13 Page 104 of 187

MAT 082 Basic Arithmetic

The Metric System

'/2 a mile

1 kilometer (km)

1 hectometer (hm)

1 dekameter (dam)

1 meter

1 decimeter (dm)

1 centimeter (cm)

1 millimeter (mm)

IM~ff!~i

= 1000 meters (m)

= 100 meters

= 10 meters

1/ 10 meter

= 1/100 meter

= 1 / 1000 meter

Page 99 of 105

i.e.

kilo hecto deka

for 1000 for 100 for 10

kilometer hectometer dekameter meter

km hm dam m

Spring 201.0

10 decimeters = 1 meter

100 centimeters = 1 meter

1000 millimeters = 1 meter

deci centi milli

for 1/10 for 1/100 for 1/100

decimeter centimeter millimeter

dm cm rnm

Converting length in the metric system:

Multzply by 10 for each step to the right --~

F- Divide by 10 for each step to the left

6.1 Conversions Homework continued: Convert as directed.

9) 8 dam tom lo) 0.37 hm tom 11) 8.64 hrn to cm 12) 0.00037 km to cm 13) 71 dm to km

English and Metric Equivalents

1 in ~ 2.54 cm

1 ft ~ 30.48 cm

1 yd ~ 0.9 rn

1 mi ~ 1.6 km

5280 ft 12 in. 2.54 cm

1 mi ~ 1 ft ~ end tin,

Unit fraction 1 inch _ 1

2.54 cm

100 miles per hour in feet per second

36 in. ~d 2.54 cm _ 1

1 yd 1 in.

6.1 Conversions Homework continued.• Convert as directed.

14) 26 in to cm 15) 26 cm to in 16) 776 mi to km 17) 776 km to mi

1$) 20 m to yd 19) 1.2 dam to in 20) 180 in to hm 21) 8 ft to m

22) 8 m to ft 23) 104 kilometers per hour in miles per hour

24) 50 miles per hour in kilometers per hour 25) 8 yd to cm 26) 1016 cm to yd

27) 50 mi to km 28) 100 miles per hour in feet per second

MCCCD/Martinez00630

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MAT 082 Basic Arithmetic Page 100 of 105 Sprang 2010

Measuring Area

Area Measures in the USA

Square Units of Measure:

1 Square foot (ftz) =144 Square inches (inZ}

1 Square yard (yd')= 9 Square feet (ftz)

1 Acre (a) = 43, 560 ftZ or 4840 ydZ

1 Square mile (miz) = 640 Acres

6.1 Conversions Homework continued: Convert as directed.

29) 20mz to ft2 30) 14 miz to km2 31) 20.6 ha to acres 32) 20 inZ to cmZ

Measuring Volume

Cubic Units USA Capacity

1 Cubic yard (yd3) ~ 200 Gallons

1 Cubic foot (ft3) ~ 7.48 Gallons

231 Cubic inches (in3) ~ 1 Gallon

Liquid and Volume Measures in the USA (ounces, pounds, tons)

USA CauacitV

1 Cup (c)

2 Pints (pt)

4 Quarts

1 Gallon

= 8 Ounces

1 Quart (qt)

= 1 Galion (gal)

= 128 Ounces (oz)

Metric Liquid: Liter

Kilo has damian Love dollars called milli.

kL hL daL L dL cL mL

kiloliter hectoliter dekaliter liter deciliter centiliter milliliter

] OOOL l 00L 1 OL 1 L o ~ L oo ~ L ~ o0o L

kg hg dag g dg cg mL

kilogram hectogram dekagram grans decigram centigram milligram

l 000g I OOg l Og 1 g 10 g oo ~ g 100o g

MCCCD/Martinez00631

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MAT 082 Basic Arithmetic Page ] O] of 105 Spring 2010

Volume —Capacity —Weight in the Metric System

Metric Liquid

Volume in Cubic Units Capacity Weight

l cm3 = 1 mL = 1 graze (g}

1 dm3 =1000 cm' = 1 L = 1 kilogram (lcg)

1 m3 = 1 kL = 1000 kilograms (kg) = 1 Tonne (t)

Measuring Weight &Temperature

Weight Measure — En lig sh SXstem SAl

16 Ounces (oz) = 1 Pound (lb)

2000 Pounds (lb) = 1 Ton (T)

Weight Measure —USA and the Metric S sY tem

1 ounce (oz) = 28 Grams (g}

1 pound (lb) = 0.45 Kilograms (kg)

1 ton (T) = 0.9 Tonne (t)

Celsius to Fahrenheit:

F=9(C~+32 5~,~~

Fahrenheit to Celsius:

C= 9 ~F-32~ 

5~7

6.1 Conversions Homework continued: Convert as directed.

33) 25,000 ft' to gal 34) 6,237 in' to gal 3S) 75,000 cm' to L

37) 16 kL to dm' 38) 6.9 dg to mg 39) 0.37kg to g

36) 4.5 L to cm'

40) 0.02m' to kg

41) 0.03k1 tog 42) 26oz to Ib 43) 220kg to lb 44) 20"C

45) 98°C 46} 86°F 47) 90°F

Selecting from millimeter, meter, and kilometer, determine the best unit of measure to express

the given length.

48. The length of a football field 49. The distance from New Yark CiTy to Washington, D.C.

50. The length of a car 51. The altitude of an airplane

52. The width of a human foot

Select the best estimate for the measure of the given item.

53. The length of this page a. 2.5 mm b. 25 mm c. 250 mm

54. The length of a pair of pants a. 700 cm b. 70 cm c. 7 cm

55. The Length of a rowboat a. 4 cm b. 4 dm c. 4 m d. 4 dam

56. The dimensions of a piece of typing paper

a. 22 mm by 28 mm b. 22 cm by 28 cm c. 22 dm by 28 dm d. 22 m by 28 m

57. If you jog six tunes axound a track that is 700 meters long, how many kilometers have you

covered?

58. The distance from New York City to Los Angeles is 4690 kilometers. What is the distance in

miles?

MCCCD/Martinez00632

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Case 2:12-cv-00702-DGC Document 68-7 Filed 10/11/13 Page 107 of 187

MAT 082 Basic Arithmetic Page 102 of 105 Spring 2010

Summation ~

This symbol: ~R x; means to add up all the x values.

The first number in the parentheses is the "x" and the second number is the "y".

Example So if we had these four pairs (-2,7), (4,5), (6,6), (7,10)

~~ x; =-2+4+6+7= 15 and ~" y~ = 7+5+6+] 0=28.

n is the number of pairs. In the example above, n is G,

Example

Use these values to do the surrunations: (0,1.1),

a)n=

b) ~

n

.x;=

C~ ~

h

; y+

(1,5.1), (2,9,3), (3,13.1), (4,16.9)

d) ~n xz; _ (this means square each x az~d then add all of the squares together)

e) ~" x; y; = (this means multiply x times y and then add all the products)

MCCCD/Martinez00633

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 240 of 243
Case 2:12-cv-00702-DGC Document 68-7 Filed 10/11/13 Page 108 of 187

MAT 082 Basic Arithmetic Page ]03 of 105 Spring 2010

~ ~

•~ a a

~ a a

~~' ~.~ ~ ]C~C~1 ~. ~ ~ ll ~ lC~l ~C ~~ ~

1~ ~ ~t~~C~]C~l

ll~~t~][~l l~]C~l~ ~

l~ ~~ 1~ ILIA' lC~~. ~ ]C~~1 ~ ] V

1~ t~ ~. 1~

Solomon Ibn Gabrio!

MCCCD/Martinez00634

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 241 of 243
Case 2:12-cv-00702-DGC Document 68-7 Filed 10/11/13 Page 109 of 187

MAT 082 Basic Arithmetic Page 104 of 1 QS Spring 2010

HOMEWORK ANSWERS FOLLOW

1.1 Numbers Homework Answers

Identify the number as Whole Number, Natural Number, Integer, Negative Number,

Positive Decimal, Negative Decimal, Improper Fraction, Proper Fraction, Mixed Number

1. 15 whole nuir~ber, natural number 2. 8 4 mixed number

3. 0 whole number 4. 3.781 positive decimal

5. 83,001 whole number, natural number 6. - 8 negative number

7. i'-~ proper fraction 8. 5 improper fraction

10. 33.7 mixed number 11. -5 negative number

12. 457 whole number, natural number 13, 5 improper fraction

14. 14 mixed number 15. -14.1 negative decimal

16. 5.8 positive decimal 17. 4 improper fraction

18. ,'-o proper fraction 19. 362,049 whole number, natural number

20. 0.1 positive decimal 21. 7 a mixed number

22. 33.7 positive decimal ~ 23. 5 improper fraction

24. 14 mixed number 25. 7r pi ~ 3.14 or ~z

26. 75,039 whole number, natural ntunber 27. 3 proper fraction

28. - 87 negative number 29. 6.49 positive fraction 30. -0.5 negative decimal

31. 2 ,o mixed number 32. '8 improper fraction 33. 14, 50, 36

MCCCD/Martinez00635

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MAT 082 Basic Arithmetic Page 1 Q5 of 105

1.2 Piace Value Homework Answers

Identify the place value of the digit 2 in each number.

1. 61,284 the 2 is in the hundreds place 2

3. 284,100 hundred thousands place 4

S. 725,837,166 ten millions place 6

7. 253,045,701,000 hundred billion place 8

Spring 2010

82,110 thousands place

823,415 ten thousands place

44,265,199 hundred thousands

823,000,419,567 ten hundred billion

Name the place value for each 0 in this number: 302,016,450,098,570

9. 1st zero is in ten trillions place 10. 4th zero is in hundred thousands place

11. 2nd zero is in hundred billions place 12. 5th zero is in the ones place

13. 3rd zero is in the one millions place

Name the place value for each 0 in this number: 810,704,069,809,Q35

14. 1st zero is in the one trillions place 15. 4th zero is in the ten thousands place

16. 2nd zero is in the ten billions place 17. 5th zero is in the hundreds place

1$. 3rd zero is in the hundred millions place

Write each number in words.

19, 8421 eight thousand four hundred twenty-one

20. 1936 one thousand nine hundred thirty-six

21. 46,205 forty-six thousand two hundred five

22. 75,089 seventy-five thousand eighty-nine

23. 3,OG4,801 three million sixty-fora thousand eight hundred one

24. 7,900,408 seven million zone hundred thousand four hundred eight

25. 46,805 26. 79,046 27. 5,600,082 28. 1,030,005

29. 571,900,000 30. 311,000,400 31. 72 32. 0, -'3 , 72, -65

33. 0, 3 , -4 5 , -'3 , 4' , 72, --65 34. 0 and 72

1.3 Rounding Numbers Homework Answers

1) SO 2) 90 3) 660 4) 120 5} 16,500 6) 12,800 7) 823,400

8) 701,500 9) 38,000 10) 117,000 71) 13,000 12) 672,000 13) 5,254,000

14) 1,396,000 15) 9,008,000 16) 3,116,000

1.4 Adding Homework Answers

1) 69 2) -103 3) 414

6) 234 7) 465 $} -507

11) -564 12) 456 13} -745

16) 7,744,824 17) 6,765,985 18) 2,874,884

20) 9,872,381 21) -4,278,923 Z2) 5,872,917

4) 38 5) 158

9) 750 10) 437

14) 6002 15) 80$

19) -6,047,114

23} 6,378,293

MCCCD/Martinez00636

Case 2:15-cv-01759-NVW Document 73-1 Filed 03/03/17 Page 243 of 243