4.4.6 Test (TST): Trigonometric Identities Test
Precalculus Sem 2 Name: Brianna
Date:
Answer the following questions using what you've learned from this unit. Write your
responses in the space provided.
1. Show that this is an identity by using the definitions of the sum and difference
formulas for cosine:
Part I: Complete the left-hand column of the table below following the steps
indicated in the right-hand column. (12 points)
Calculation Reason
Given on the right
side of the original
problem
½ (cos a cos b + sin a sin b- cos a
cos b- sin a sinb)
                                           
Apply the definitions
of the sum and
difference identities
for cosine
- Cos(a) cos(b)/2
                                          
Simplify the
expressions
2. Use the subtraction formula for sine to verify this identity:
Part I: Complete the left-hand column of the table below following the steps
indicated in the right-hand column. (12 points)
Calculation Reason
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Given on the left
side of the original
problem
cosx
                              
             
Apply the
difference identity
for sine
cosx
                              
             
Simplify the
expressions
3. Find the exact value of each of the following. Do not use a calculator —
answers given as decimals will not receive credit.
   A.
Part I: square root(2)- square root(6)/4
Find two common angles that either add up to or differ by . Rewrite this
problem as the cosine of either a sum or a difference of those two angles. (2
points)
Cos pi/3 + pi/4
Part II:
Evaluate the expression. (4 points)
(square root of 2 minus the square root of 6)/4
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   B.
Part I:
Find two common angles that either add up to or differ by . Rewrite this
problem as the tangent of either a sum or a difference of those two angles. (2
points)
2pi/3 + 3pi/4
Part II:
Evaluate the expression. (4 points)
Square root 3 + 2
   C.
Part I:
Find two common angles that either add up to or differ by . Rewrite this
problem as the sine of either a sum or a difference of those two angles. (2
points)
sin ( 135+ 60)
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Part II:
Evaluate the expression. (4 points)
Squae root 2 minus square root 6/4
   D.
Part I:
Find two common angles that differ by . Rewrite this problem as the
cotangent of a difference of those two angles. (2 points)
cot(45-30)
Part II:
Evaluate the expression. (4 points)
Square root 3+2
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4. Rewrite so that it involves only the first power of cosine.
Part I: Complete the left-hand column of the table below following the steps
indicated in the right-hand column. (10 points)
Calculation Reason
Given in the problem
Sin²x * sin²x
                                           
Rewrite the power as a product of
two squared terms
(1-cos²x)(1-cos²x) Apply the squared identity for sine
twice
(1+cosx)(1-cosx)(1+cos
x)(1-cosx)
                                           
Multiply the terms
(1+cosx)²(1-cosx)²
                                           
Apply the squared identity for
cosine
1−2cos²(x)+cos⁴(x)
Simplify the expression
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5. Prove the identity:
Part I: Complete the left-hand column of the table below following the steps
indicated in the right-hand column. (12 points)
Calculation Reason
Given on the
left side of the
original
problem
2sinxcosx/sinx-cos²
x-sin²x/cosx
                                     
      
Apply the
double-angle
formulas for
sine and cosine
sec(x)
                                     
      
Simplify the
expression
In a right triangle,
the secant of an
angle is the length of
the hypotenuse
divided by the kngth
of the adjacent side
                                     
      
Definition of
secant
6. Prove the identity:
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Part I: Complete the left-hand column of the table below following the steps
indicated in the right-hand column. (12 points)
Calculation Reason
Given on
the left
side of
the
original
problem
(cosx²+cosy²)(cos²x+cos²
y)+(sin²-sin²y)(sin²x-sin²y)
                                           
Expand
each
squared
term
2cos(x)cos(y)+2-2sin x sin
y
                                           
Simplify
the
expressio
n
sin²(a)+cos²(a)=1
                                           
Apply the
Pythagor
ean
identity
2+2cos (x + y)
Apply the
addition
formula
for
cosine
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7. Solve these equations:
   A.
Part I:
Apply the sum and difference formulas for sine to simplify the left-hand side
of the equation. (2 points)
cos x- cos x=1
Part II:
Find all solutions to the equation you found in part I. (4 points)
x=pi/4 +2pi n, 3pi/4 +2pi n
   B.
Part I:
Use the double-angle formula for sine to rewrite the problem in terms of sine
only. (2 points)
sin2x=2sin
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Part II:
Solve the equation, finding all solutions. (4 points)
x=pi/6 and pi/3
   C.
Part I:
Factor the expression in the equation to make it easier to solve. (2 points)
(tan x +2)(tan x =1)=0
Part II:
Solve the equation, finding all solutions. (4 points)
x= 1.107 + pi n
x= pi/4