trigonometry

trigonometry

1

Geometrical definitions

Trigonometry

arose in ancient times out of attempts to measure the lengths of various lines associated with circles. For example, consider this diagram showing the upper right

quadrant

with a

unit circle

:

The point

B

, external to the circle with center

O

, determines a unique secant

B

⁢

O

and two

tangent lines

of equal length, one of which is

B

⁢

A

. It thus also uniquely determines an angle

α

=

∠

⁢

A

⁢

O

⁢

B

. The

trigonometric functions

sec

and

tan

simply measure the ratio of the lengths of the secant and tangent lines to the radius of the circle.

Thus, in the diagram,

tan

⁡

α

=

A

⁢

B

=

A

⁢

B

:

O

⁢

A

. But

triangles

O

⁢

A

⁢

B

and

O

⁢

D

⁢

C

are obviously

similar

, so

tan

⁡

α

is also equal to

C

⁢

D

:

O

⁢

D

, and we recover the usual current-day definition of

tan

. Similarly, from the diagram,

sec

⁡

α

=

O

⁢

B

=

O

⁢

B

:

O

⁢

A

and again, using

similarity

, we also have

sec

⁡

α

=

O

⁢

C

:

O

⁢

D

, which is again the usual modern definition.

The

cosecant

,

csc

, is the ratio of the secant to the

other

leg of the

right triangle

. Using the standard definitions today for

sec

and

csc

, we see that this is correct:

csc

⁡

(

x

)

=

sec

⁡

(

x

)

tan

⁡

(

x

)

=

1

cos

⁡

(

x

)

⋅

c

⁢

o

⁢

s

⁢

(

x

)

sin

⁡

(

x

)

=

1

sin

⁡

(

x

)

The sine arose in attempts to measure the length of a chord traversing a given angle on a circle. Consider the following diagram:

Here, the arc

C

⁢

E

is the circular angle

α

, which is also

∠

⁢

C

⁢

F

⁢

E

,

∠

⁢

C

⁢

O

⁢

D

, and

∠

⁢

D

⁢

O

⁢

E

(recall that a

circumferential angle

is the same measure as the arc subtended, while a

central angle

is twice the arc subtended). The length of the chord

C

⁢

E

is in fact

2

⁢

sin

⁡

α

. Presumably because central angles are easier to work with, the common definition was that of the half-chord, and one can see by considering

△

⁢

C

⁢

O

⁢

D

that in fact

C

⁢

D

is

sin

⁡

α

using the modern definitions.

The term “sine” has an interesting history. The Hindus gave the name

jiva

to the half-chord

C

⁢

D

; the Arabs used (or created) the word

jiba

for this concept. When Robert of Chester, an early translator of al-Khowarizmi’s

Algebra

, translated this word, he mistook it for a similar Arabic word,

jaib

, which means “bay” or “inlet”. As a result, he used the Latin term “sinus”, which also means bay or inlet.

To understand how the trigonometric functions are defined today, consider the right triangle

A

⁢

B

⁢

C

below.

Noting that

0

∘

<

α

<

90

∘

, we define the trigonometric functions

sine, cosine, tangent, secant, cosecant and

cotangent

for angle

α

respectively as:

sin

⁡

(

α

)

=

B

⁢

C

C

⁢

A

,

csc

⁡

(

α

)

=

C

⁢

A

B

⁢

C

cos

⁡

(

α

)

=

A

⁢

B

C

⁢

A

,

sec

⁡

(

α

)

=

C

⁢

A

A

⁢

B

tan

⁡

(

α

)

=

B

⁢

C

A

⁢

B

,

cot

⁡

(

α

)

=

A

⁢

B

B

⁢

C

We will discuss later how to extend these definitions to a broader set of values for

α

.

Several identities follow directly from the definitions:

•

tan

⁡

(

α

)

=

sin

⁡

(

α

)

/

cos

⁡

(

α

)

.

•

sec

⁡

(

α

)

=

1

/

cos

⁡

(

α

)

,

csc

⁡

(

α

)

=

1

/

sin

⁡

(

α

)

,

cot

⁡

(

α

)

=

1

/

tan

⁡

(

α

)

.

•

sin

⁡

(

α

)

=

cos

⁡

(

90

∘

-

α

)

,

tan

⁡

(

α

)

=

cot

⁡

(

90

∘

-

α

)

,

sec

⁡

(

α

)

=

csc

⁡

(

90

∘

-

α

)

.

The last property follows from the fact that

∠

⁢

A

⁢

C

⁢

B

=

90

∘

-

α

.

The

Pythagorean theorem

states that

C

⁢

A

2

=

A

⁢

B

2

+

B

⁢

C

2

and thus

1

=

C

⁢

A

2

C

⁢

A

2

=

A

⁢

B

2

+

B

⁢

C

2

C

⁢

A

2

=

(

A

⁢

B

C

⁢

A

)

2

+

(

B

⁢

C

C

⁢

A

)

2

=

(

cos

⁡

(

α

)

)

2

+

(

sin

⁡

(

α

)

)

2

.

It is customary to write

(

sin

⁡

(

α

)

)

n

,

(

cos

⁡

(

α

)

)

n

, etc. as

sin

n

⁡

(

α

)

,

cos

n

⁡

(

α

)

, etc. respectively, so the previous identity is usually written as

sin

2

⁡

(

α

)

+

cos

2

⁡

(

α

)

=

1

and it is known as the

Pythagorean identity

.
Notice that the first three identities let us to express any expression involving trigonometric functions using only sines and cosines, whereas the Pythagorean identity lets us reduce it further, using only sines. This technique is sometimes used when proving trigonometric identities.

Two other identities that can be obtained from the Pythagorean theorem or from the Pythagorean identity are

tan

2

⁡

(

α

)

+

1

=

sec

2

⁡

(

α

)

,

1

+

cot

2

⁡

(

α

)

=

csc

2

⁡

(

α

)

.

2

Extending the domain

There are several approaches for extending the domain of the trigonometric functions so they are not restricted to angles between

0

and

90

∘

. We could use the

angle sum identities

below in order to calculate the value of the trigonometric functions outside the given range (for instance,

sin

⁡

(

120

∘

)

could be found as

sin

⁡

(

60

∘

+

60

∘

)

. Some other more formal approaches use power series

expansion

to define the

functions

for any real value (or even complex!), but we will use a more geometrical approach, using the unit circle.

Consider a unit circle centered at origin of the plane (that is, a circle of radius

1

and center

(

0

,

0

)

), and draw a ray from the center making an angle of

α

with the horizontal axis, measured counter-clockwise from the positive

x

-axis. (This ray is the

terminal ray

of the angle.) Let

P

be the point of

intersection

of this line with the circle, and

T

the intersection with the line

x

=

1

. Finally, let us denote by

(

x

,

y

)

the

coordinates

of

P

and

(

1

,

t

)

the coordinates of

T

.

Since the circle has radius

1

, the

distance

from origin to

P

is also

1

and thus

cos

⁡

α

=

x

1

=

x

,

sin

⁡

α

=

y

1

=

y

,

in other words, the coordinates of

P

are precisely

(

cos

⁡

α

,

sin

⁡

α

)

.

Now, the two right triangles made with the

projections

of

P

and

T

to the

x

-axis are similar, so we have

tan

⁡

α

=

y

x

=

t

1

=

t

,

that is, the point

t

has

ordinate

equal to

tan

⁡

α

.

This formulation makes it much easier to generalize the definitions to arbitrary angles:

cos

⁡

α

,

sin

⁡

α

are the coordinates of the intersection of the circle with the line through the origin making angle

α

with the positive

x

-axis:

You may be wondering now why we put

-

tan

⁡

α

instead of

tan

⁡

α

if the analogy was to be continued. The reason is that we only need

sin

⁡

α

and

cos

⁡

α

to define all the other trigonometric functions, and given that

tan

⁡

α

=

sin

⁡

α

cos

⁡

α

we use that identity to extend the domain of

tan

⁡

α

. In the particular case of the drawing above, both

cos

⁡

α

and

sin

⁡

α

are negative, so their quotient must be positive, and that’s why we added a minus sign to

tan

⁡

α

.

The geometrical approach lets us easily verify the following relations:

sin

⁡

(

180

∘

-

α

)

=

sin

⁡

α

,

sin

⁡

(

180

∘

+

α

)

=

-

sin

⁡

α

,

sin

⁡

(

-

α

)

=

-

sin

⁡

α

cos

⁡

(

180

∘

-

α

)

=

-

cos

⁡

α

,

cos

⁡

(

180

∘

+

α

)

=

-

cos

⁡

α

,

cos

⁡

(

-

α

)

=

cos

⁡

α

tan

⁡

(

180

∘

-

α

)

=

-

tan

⁡

α

,

tan

⁡

(

180

∘

+

α

)

=

tan

⁡

α

,

tan

⁡

(

-

α

)

=

-

tan

⁡

α

3

Graphs

The graph of

sin

appears below. Note that the

x

-axis is scaled not in degrees, but rather in radians.

The graph of

cos

is below. Note the similarity between the two graphs: the graphs of

cos

is just like the graph of

sin

, but shifted by

90

∘

(

π

/

2

radians). This precisely

reflects

the identity

sin

⁡

α

=

cos

⁡

(

α

-

90

∘

)

.

Also notice that you can read off directly from the graphs the facts that

sin

⁡

α

=

-

sin

⁡

(

-

α

)

,

cos

⁡

α

=

cos

⁡

(

-

α

)

4

Trigonometrical identities

There are literally thousands of trigonometric identities (

http://planetmath.org/GoniometricFormulae

). Some of the most common (and most useful) are:

Sum and difference of angles

sin

⁡

(

x

+

y

)

=

sin

⁡

x

⁢

cos

⁡

y

+

cos

⁡

x

⁢

sin

⁡

y

,

⁢

sin

⁡

(

x

-

y

)

=

sin

⁡

x

⁢

cos

⁡

y

-

cos

⁡

x

⁢

sin

⁡

y

,

cos

⁡

(

x

+

y

)

=

cos

⁡

x

⁢

cos

⁡

y

-

sin

⁡

x

⁢

sin

⁡

y

,

⁢

cos

⁡

(

x

-

y

)

=

cos

⁡

x

⁢

cos

⁡

y

+

sin

⁡

x

⁢

sin

⁡

y

,

tan

⁡

(

x

+

y

)

=

tan

⁡

x

+

tan

⁡

y

1

-

tan

⁡

x

⁢

tan

⁡

y

,

⁢

tan

⁡

(

x

-

y

)

=

tan

⁡

x

-

tan

⁡

y

1

+

tan

⁡

x

⁢

tan

⁡

y

.

(There is a

proof of angle sum identities

, as well as a

geometric derivation of addition formulas for sine and cosine

, on this site).

Half and double angles

The

double angle formulas

are derived directly from the sum of angles formulas above. The

half-angle formulas

can then be derived from the double angle formulas by substituting

x

/

2

for

x

and simplifying, using the identity

s

⁢

i

⁢

n

2

+

c

⁢

o

⁢

s

2

=

1

.

sin

⁡

(

x

/

2

)

=

±

1

-

cos

⁡

x

2

,

sin

⁡

(

2

⁢

x

)

=

2

⁢

sin

⁡

x

⁢

cos

⁡

x

,

cos

⁡

(

x

/

2

)

=

±

1

+

cos

⁡

x

2

,

cos

⁡

(

2

⁢

x

)

=

cos

2

⁡

x

-

sin

2

⁡

x

=

2

⁢

cos

2

⁡

x

-

1

=

1

-

2

⁢

sin

2

⁡

x

.

Title

trigonometry

Canonical name

Trigonometry

Date of creation

2013-03-22 15:05:00

Last modified on

2013-03-22 15:05:00

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

31

Author

rm50 (10146)

Entry type

Topic

Classification

msc 43-00

Classification

msc 51-00

Classification

msc 42-00

Classification

msc 33B10

Related topic

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Related topic

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Related topic

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Related topic

GoniometricFormulae

Related topic

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Related topic

CyclometricFunctions

Defines

sine

Defines

cosine

Defines

tangent

Defines

secant

Defines

cosecant

Defines

cotangent

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