Question 1

Express the sum or difference as a product of sines and/or cosines.
-$\sin ( 7 \theta ) + \sin ( 3 \theta )$

Accepted Answer

A)  $2 \sin ( 5 \theta ) \sin ( 2 \theta )$ 
B)  $2 \cos ( 5 \theta ) \sin ( 2 \theta )$ 
C)  $2 \sin ( 10 \theta )$ 
D)  $2 \sin ( 5 \theta ) \cos ( 2 \theta )$ 
A)  $2 \sin ( 5 \theta ) \sin ( 2 \theta )$ 
B)  $2 \cos ( 5 \theta ) \sin ( 2 \theta )$ 
C)  $2 \sin ( 10 \theta )$ 
D)  $2 \sin ( 5 \theta ) \cos ( 2 \theta )$

Question 2

Complete the identity.
-$\frac { \sin ( 2 \theta ) + \sin ( 4 \theta ) } { \cos ( 5 \theta ) + \cos ( 7 \theta ) } = ?$

Accepted Answer

A)  $\frac { \cos ( 6 \theta ) } { \sin ( 12 \theta ) }$ 
B)  $\frac { \sin ( 3 \theta ) } { \cos ( 6 \theta ) }$ 
C)  $\tan ( 3 \theta ) + \tan ( 6 \theta )$ 
D)  $\frac { \sin ( 6 \theta ) } { \cos ( 12 \theta ) }$ 
A)  $\frac { \cos ( 6 \theta ) } { \sin ( 12 \theta ) }$ 
B)  $\frac { \sin ( 3 \theta ) } { \cos ( 6 \theta ) }$ 
C)  $\tan ( 3 \theta ) + \tan ( 6 \theta )$ 
D)  $\frac { \sin ( 6 \theta ) } { \cos ( 12 \theta ) }$

Question 3

Solve the equation on the interval $$0 \leq \theta < 2 \pi$$
-$\sin ^ { 2 } ( 2 \theta ) = 1$

Accepted Answer

A)  $0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
B)  $\frac { \pi } { 8 } , \frac { 9 \pi } { 8 }$ 
C)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
D)  No solution 
A)  $0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
B)  $\frac { \pi } { 8 } , \frac { 9 \pi } { 8 }$ 
C)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
D)  No solution

Question 4

Use the Half-angle Formulas to find the exact value of the trigonometric function.
-$\sin \frac { 5 \pi } { 12 }$

Accepted Answer

A)  $- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
B)  $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
C)  $\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
D)  $- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
A)  $- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
B)  $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
C)  $\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
D)  $- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$

Question 5

Find the exact value of the expression.
-$$\tan ^ { - 1 } \frac { \sqrt { 3 } } { 3 }$$

Accepted Answer

A)  $\frac { \pi } { 6 }$ 
B)  $\frac { \pi } { 4 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { 7 \pi } { 6 }$ 
A)  $\frac { \pi } { 6 }$ 
B)  $\frac { \pi } { 4 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { 7 \pi } { 6 }$

Question 6

Find the exact value of the expression.
-$$\cos ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 2 } \right)$$

Accepted Answer

A)  $\frac { 5 \pi } { 6 }$ 
B)  $\frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $\frac { 2 \pi } { 3 }$ 
A)  $\frac { 5 \pi } { 6 }$ 
B)  $\frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $\frac { 2 \pi } { 3 }$

Question 7

Write the trigonometric expression as an algebraic expression in u.
-$\cos \left( \sin ^ { - 1 } u \right)$

Accepted Answer

A)  $\sqrt { \mathrm { u } ^ { 2 } + 1 }$ 
B)  $\sqrt { 1 - u ^ { 2 } }$ 
C)  $\sqrt { \mathrm { u } ^ { 2 } - 1 }$ 
D)  $\frac { \sqrt { \mathrm { u } ^ { 2 } + 1 } } { \mathrm { u } }$ 
A)  $\sqrt { \mathrm { u } ^ { 2 } + 1 }$ 
B)  $\sqrt { 1 - u ^ { 2 } }$ 
C)  $\sqrt { \mathrm { u } ^ { 2 } - 1 }$ 
D)  $\frac { \sqrt { \mathrm { u } ^ { 2 } + 1 } } { \mathrm { u } }$

Question 8

Establish the identity.
-$$\frac { \csc x - 1 } { \csc x + 1 } = \frac { \cot ^ { 2 } x } { \csc ^ { 2 } x + 2 \csc x + 1 }$$

Accepted Answer

The answer of Establish the identity.
-\[\frac { \csc x -...

Question 9

Complete the identity.
-$$\frac { \sin \theta } { 1 + \sin \theta } - \frac { \sin \theta } { 1 - \sin \theta } = ?$$

Accepted Answer

A)  $\sin \theta \tan \theta$ 
B)  $1 + \cot \theta$ 
C)  $\sec \theta \csc \theta$ 
D)  $- 2 \tan ^ { 2 } \theta$ 
A)  $\sin \theta \tan \theta$ 
B)  $1 + \cot \theta$ 
C)  $\sec \theta \csc \theta$ 
D)  $- 2 \tan ^ { 2 } \theta$

Question 10

Complete the identity.
-$$\cos ( 4 \theta ) = \text { ? }$$

Accepted Answer

A)  $3 \sin \theta - 4 \sin ^ { 3 } \theta$ 
B)  $\cos ^ { 4 } \theta - 6 \sin ^ { 2 } \theta \cos ^ { 2 } \theta + \sin ^ { 4 } \theta$ 
C)  $4 \sin \theta \cos ^ { 3 } \theta - 4 \sin ^ { 3 } \theta \cos \theta$ 
D)  $\cos ^ { 3 } \theta - 3 \sin ^ { 2 } \theta \cos \theta$ 
A)  $3 \sin \theta - 4 \sin ^ { 3 } \theta$ 
B)  $\cos ^ { 4 } \theta - 6 \sin ^ { 2 } \theta \cos ^ { 2 } \theta + \sin ^ { 4 } \theta$ 
C)  $4 \sin \theta \cos ^ { 3 } \theta - 4 \sin ^ { 3 } \theta \cos \theta$ 
D)  $\cos ^ { 3 } \theta - 3 \sin ^ { 2 } \theta \cos \theta$

Question 11

Complete the identity.
-$\sec ^ { 4 } \theta - 2 \sec ^ { 2 } \theta \tan ^ { 2 } \theta + \tan ^ { 4 } \theta =$ ?

Accepted Answer

A)  2 
B)  $\sec ^ { 2 } \theta \left( 1 + \tan ^ { 2 } \theta \right)$ 
C)  1 
D)  $\sec ^ { 2 } \theta + \tan ^ { 2 } \theta$ 
A)  2 
B)  $\sec ^ { 2 } \theta \left( 1 + \tan ^ { 2 } \theta \right)$ 
C)  1 
D)  $\sec ^ { 2 } \theta + \tan ^ { 2 } \theta$

Question 12

Find the exact value of the expression.
-$\cos ^ { - 1 } \left( \cos \frac { 7 \pi } { 6 } \right)$

Accepted Answer

A)  $\frac { 4 \pi } { 5 }$ 
B)  $\frac { \pi } { 6 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { 5 \pi } { 6 }$ 
A)  $\frac { 4 \pi } { 5 }$ 
B)  $\frac { \pi } { 6 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { 5 \pi } { 6 }$

Question 13

Complete the identity.
-$$( 4 \sin \theta \cos \theta ) \left( 1 - 2 \sin ^ { 2 } \theta \right) = \text { ? }$$

Accepted Answer

A)  $\sin ( 4 \theta )$ 
B)  $\sin ( 2 \theta ) + \cos ( 2 \theta )$ 
C)  $\frac { \cos ( 4 \theta ) } { \sin ( 2 \theta ) }$ 
D)  $\sin ( 2 \theta ) \cos ( 2 \theta )$ 
A)  $\sin ( 4 \theta )$ 
B)  $\sin ( 2 \theta ) + \cos ( 2 \theta )$ 
C)  $\frac { \cos ( 4 \theta ) } { \sin ( 2 \theta ) }$ 
D)  $\sin ( 2 \theta ) \cos ( 2 \theta )$

Question 14

Use a calculator to solve the equation on the interval 0 $0 \leq \theta < 2 \pi$ . Round the answer to two decimal places.
-$\cos \theta = - 0.55$

Accepted Answer

A)  2.15, 4.13 
B)  0.99, 2.15 
C)  0.99, 4.13 
D)  2.15, 5.29 
A)  2.15, 4.13 
B)  0.99, 2.15 
C)  0.99, 4.13 
D)  2.15, 5.29

Question 15

Complete the identity.
-$\sin ( \alpha + \beta ) \cos ( \alpha - \beta ) =$ ?

Accepted Answer

A)  $\sin \alpha \cos \alpha - \sin \alpha \cos \beta + \sin \beta \cos \alpha + \sin \beta \cos \beta$ 
B)  $\sin ^ { 2 } \alpha \cos ^ { 2 } \beta - \sin ^ { 2 } \beta \cos ^ { 2 } \alpha$ 
C)  $\sin \alpha \cos \alpha + \sin \beta \cos \beta$ 
D)  $\sin ^ { 2 } \alpha - \cos ^ { 2 } \beta$ 
A)  $\sin \alpha \cos \alpha - \sin \alpha \cos \beta + \sin \beta \cos \alpha + \sin \beta \cos \beta$ 
B)  $\sin ^ { 2 } \alpha \cos ^ { 2 } \beta - \sin ^ { 2 } \beta \cos ^ { 2 } \alpha$ 
C)  $\sin \alpha \cos \alpha + \sin \beta \cos \beta$ 
D)  $\sin ^ { 2 } \alpha - \cos ^ { 2 } \beta$

Question 16

Complete the identity.
-$\sin ( 4 \theta ) \sin ( 7 \theta ) \cos ( 4 \theta ) \cos ( 7 \theta ) = ?$

Accepted Answer

A)  $\cos ^ { 2 } ( 56 \theta )$ 
B)  $\frac { \sin ^ { 2 } ( 56 \theta ) } { 4 }$ 
C)  $\frac { \cos ^ { 2 } ( 11 \theta ) + \cos ^ { 2 } ( 3 \theta ) } { 4 }$ 
D)  $\frac { \cos ^ { 2 } ( 3 \theta ) - \cos ^ { 2 } ( 11 \theta ) } { 4 }$ 
A)  $\cos ^ { 2 } ( 56 \theta )$ 
B)  $\frac { \sin ^ { 2 } ( 56 \theta ) } { 4 }$ 
C)  $\frac { \cos ^ { 2 } ( 11 \theta ) + \cos ^ { 2 } ( 3 \theta ) } { 4 }$ 
D)  $\frac { \cos ^ { 2 } ( 3 \theta ) - \cos ^ { 2 } ( 11 \theta ) } { 4 }$

Question 17

Establish the identity.
-$$\cot ^ { 2 } x + \csc ^ { 2 } x = 2 \csc ^ { 2 } x - 1$$

Accepted Answer

The answer of Establish the identity.
-\[\cot ^ { 2 }...

Question 18

Express the product as a sum containing only sines or cosines.
-$- 2 \sin ( 5 \theta ) \sin \theta$

Accepted Answer

A)  $\cos ( 7 \theta ) - \cos ( 3 \theta )$ 
B)  $\cos ( 6 \theta ) + \cos ( 4 \theta )$ 
C)  $\cos ( 7 \theta ) + \cos ( 3 \theta )$ 
D)  $\cos ( 6 \theta ) - \cos ( 4 \theta )$ 
A)  $\cos ( 7 \theta ) - \cos ( 3 \theta )$ 
B)  $\cos ( 6 \theta ) + \cos ( 4 \theta )$ 
C)  $\cos ( 7 \theta ) + \cos ( 3 \theta )$ 
D)  $\cos ( 6 \theta ) - \cos ( 4 \theta )$

Question 19

Complete the identity.
-$\tan \theta ( \cot \theta - \cos \theta ) =$ ?

Accepted Answer

A)  1 
B)  $1 - \sin \theta$ 
C)  $- \sec ^ { 2 } \theta$ 
D)  0 
A)  1 
B)  $1 - \sin \theta$ 
C)  $- \sec ^ { 2 } \theta$ 
D)  0

Question 20

Solve the problem.
-Find $\cos \theta$, given that $\cos 2 \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in quadrant I.

Accepted Answer

A)  $\frac { \sqrt { 8 - 2 \sqrt { 10 } } } { 4 }$ 
B)  $\frac { \sqrt { 6 } } { 4 }$ 
C)  $\frac { \sqrt { 10 } } { 4 }$ 
D)  $\frac { \sqrt { 8 - 2 \sqrt { 5 } } } { 2 }$ 
A)  $\frac { \sqrt { 8 - 2 \sqrt { 10 } } } { 4 }$ 
B)  $\frac { \sqrt { 6 } } { 4 }$ 
C)  $\frac { \sqrt { 10 } } { 4 }$ 
D)  $\frac { \sqrt { 8 - 2 \sqrt { 5 } } } { 2 }$

Express the sum or difference as a product of sines and/or cosines. - $\sin ( 7 \theta ) + \sin ( 3 \theta )$

Complete the identity. - $\frac { \sin ( 2 \theta ) + \sin ( 4 \theta ) } { \cos ( 5 \theta ) + \cos ( 7 \theta ) } = ?$

Solve the equation on the interval $0 \leq \theta < 2 \pi$ - $\sin ^ { 2 } ( 2 \theta ) = 1$

Use the Half-angle Formulas to find the exact value of the trigonometric function. - $\sin \frac { 5 \pi } { 12 }$

Find the exact value of the expression. - $\tan ^ { - 1 } \frac { \sqrt { 3 } } { 3 }$

Find the exact value of the expression. - $\cos ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 2 } \right)$

Write the trigonometric expression as an algebraic expression in u. - $\cos \left( \sin ^ { - 1 } u \right)$

Establish the identity. - $\frac { \csc x - 1 } { \csc x + 1 } = \frac { \cot ^ { 2 } x } { \csc ^ { 2 } x + 2 \csc x + 1 }$

Complete the identity. - $\frac { \sin \theta } { 1 + \sin \theta } - \frac { \sin \theta } { 1 - \sin \theta } = ?$

Complete the identity. - $\cos ( 4 \theta ) = \text { ? }$

Complete the identity. - $\sec ^ { 4 } \theta - 2 \sec ^ { 2 } \theta \tan ^ { 2 } \theta + \tan ^ { 4 } \theta =$ ?

Find the exact value of the expression. - $\cos ^ { - 1 } \left( \cos \frac { 7 \pi } { 6 } \right)$

Complete the identity. - $( 4 \sin \theta \cos \theta ) \left( 1 - 2 \sin ^ { 2 } \theta \right) = \text { ? }$

Use a calculator to solve the equation on the interval 0 $0 \leq \theta < 2 \pi$ . Round the answer to two decimal places. - $\cos \theta = - 0.55$

Complete the identity. - $\sin ( \alpha + \beta ) \cos ( \alpha - \beta ) =$ ?

Complete the identity. - $\sin ( 4 \theta ) \sin ( 7 \theta ) \cos ( 4 \theta ) \cos ( 7 \theta ) = ?$

Establish the identity. - $\cot ^ { 2 } x + \csc ^ { 2 } x = 2 \csc ^ { 2 } x - 1$

Express the product as a sum containing only sines or cosines. - $- 2 \sin ( 5 \theta ) \sin \theta$

Complete the identity. - $\tan \theta ( \cot \theta - \cos \theta ) =$ ?

Solve the problem. -Find $\cos \theta$ , given that $\cos 2 \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in quadrant I.

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Exam 6: Analytic Trigonometry

Express the sum or difference as a product of sines and/or cosines. - sin⁡(7θ)+sin⁡(3θ)\sin ( 7 \theta ) + \sin ( 3 \theta )sin(7θ)+sin(3θ)

Complete the identity. - sin⁡(2θ)+sin⁡(4θ)cos⁡(5θ)+cos⁡(7θ)=?\frac { \sin ( 2 \theta ) + \sin ( 4 \theta ) } { \cos ( 5 \theta ) + \cos ( 7 \theta ) } = ?cos(5θ)+cos(7θ)sin(2θ)+sin(4θ)​=?

Solve the equation on the interval 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π - sin⁡2(2θ)=1\sin ^ { 2 } ( 2 \theta ) = 1sin2(2θ)=1

Use the Half-angle Formulas to find the exact value of the trigonometric function. - sin⁡5π12\sin \frac { 5 \pi } { 12 }sin125π​

Find the exact value of the expression. - tan⁡−133\tan ^ { - 1 } \frac { \sqrt { 3 } } { 3 }tan−133​​

Find the exact value of the expression. - cos⁡−1(−32)\cos ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 2 } \right)cos−1(−23​​)

Write the trigonometric expression as an algebraic expression in u. - cos⁡(sin⁡−1u)\cos \left( \sin ^ { - 1 } u \right)cos(sin−1u)

Establish the identity. - csc⁡x−1csc⁡x+1=cot⁡2xcsc⁡2x+2csc⁡x+1\frac { \csc x - 1 } { \csc x + 1 } = \frac { \cot ^ { 2 } x } { \csc ^ { 2 } x + 2 \csc x + 1 }cscx+1cscx−1​=csc2x+2cscx+1cot2x​

Complete the identity. - sin⁡θ1+sin⁡θ−sin⁡θ1−sin⁡θ=?\frac { \sin \theta } { 1 + \sin \theta } - \frac { \sin \theta } { 1 - \sin \theta } = ?1+sinθsinθ​−1−sinθsinθ​=?

Complete the identity. - cos⁡(4θ)= ? \cos ( 4 \theta ) = \text { ? }cos(4θ)= ?

Complete the identity. - sec⁡4θ−2sec⁡2θtan⁡2θ+tan⁡4θ=\sec ^ { 4 } \theta - 2 \sec ^ { 2 } \theta \tan ^ { 2 } \theta + \tan ^ { 4 } \theta =sec4θ−2sec2θtan2θ+tan4θ= ?

Find the exact value of the expression. - cos⁡−1(cos⁡7π6)\cos ^ { - 1 } \left( \cos \frac { 7 \pi } { 6 } \right)cos−1(cos67π​)

Complete the identity. - (4sin⁡θcos⁡θ)(1−2sin⁡2θ)= ? ( 4 \sin \theta \cos \theta ) \left( 1 - 2 \sin ^ { 2 } \theta \right) = \text { ? }(4sinθcosθ)(1−2sin2θ)= ?

Use a calculator to solve the equation on the interval 0 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π . Round the answer to two decimal places. - cos⁡θ=−0.55\cos \theta = - 0.55cosθ=−0.55

Complete the identity. - sin⁡(α+β)cos⁡(α−β)=\sin ( \alpha + \beta ) \cos ( \alpha - \beta ) =sin(α+β)cos(α−β)= ?

Complete the identity. - sin⁡(4θ)sin⁡(7θ)cos⁡(4θ)cos⁡(7θ)=?\sin ( 4 \theta ) \sin ( 7 \theta ) \cos ( 4 \theta ) \cos ( 7 \theta ) = ?sin(4θ)sin(7θ)cos(4θ)cos(7θ)=?

Establish the identity. - cot⁡2x+csc⁡2x=2csc⁡2x−1\cot ^ { 2 } x + \csc ^ { 2 } x = 2 \csc ^ { 2 } x - 1cot2x+csc2x=2csc2x−1

Express the product as a sum containing only sines or cosines. - −2sin⁡(5θ)sin⁡θ- 2 \sin ( 5 \theta ) \sin \theta−2sin(5θ)sinθ

Complete the identity. - tan⁡θ(cot⁡θ−cos⁡θ)=\tan \theta ( \cot \theta - \cos \theta ) =tanθ(cotθ−cosθ)= ?

Solve the problem. -Find cos⁡θ\cos \thetacosθ , given that cos⁡2θ=14\cos 2 \theta = \frac { 1 } { 4 }cos2θ=41​ and θ\thetaθ terminates in quadrant I.