Question 1

Express the sum or difference as a product of sines and/or cosines.
-$\sin ( 6 \theta ) - \sin ( 4 \theta )$

Accepted Answer

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

Question 2

Solve the problem using Snell's Law: $$\frac { \sin \theta _ { 1 } } { \sin \theta _ { 2 } } = \frac { v _ { 1 } } { v _ { 2 } }$$
-A ray of light near the horizon with an angle of incidence of $77 ^ { \circ }$ enters a pool of water and strikes a fish's eye. If the index of refraction is $1.33$, what is the angle of refraction (to two decimal places)?

Accepted Answer

A)  $42.89 ^ { \circ }$ 
B)  $46.20 ^ { \circ }$ 
C)  $47.11 ^ { \circ }$ 
D)  $43.80 ^ { \circ }$ 
A)  $42.89 ^ { \circ }$ 
B)  $46.20 ^ { \circ }$ 
C)  $47.11 ^ { \circ }$ 
D)  $43.80 ^ { \circ }$ C

Question 3

Choose the one alternative that best completes the statement or answers the question.
Find the exact value of the expression.
-$$\cos \left( 2 \sin ^ { - 1 } \frac { 1 } { 4 } \right)$$

Accepted Answer

A)  $\frac { 3 } { 8 }$ 
B)  $\frac { 5 } { 8 }$ 
C)  $\frac { 1 } { 8 }$ 
D)  $\frac { 7 } { 8 }$ 
A)  $\frac { 3 } { 8 }$ 
B)  $\frac { 5 } { 8 }$ 
C)  $\frac { 1 } { 8 }$ 
D)  $\frac { 7 } { 8 }$ D

Question 4

Use the Half-angle Formulas to find the exact value of the trigonometric function.
-$\cos 75 ^ { \circ }$

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

Choose the one alternative that best completes the statement or answers the question.
Complete the identity.
-$\sec \theta - \frac { 1 } { \sec \theta } =$ ?

Accepted Answer

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

Question 6

Solve the equation. Give a general formula for all the solutions.
-$$\cos \theta = 1$$

Accepted Answer

A)  $\theta = \frac { 3 \pi } { 2 } + 2 \mathrm { k } \pi$ 
B)  $\theta = \frac { \pi } { 2 } + 2 k \pi$ 
C)  $\theta = \pi + 2 k \pi$ 
D)  $\theta = 0 + 2 \mathrm { k } \pi$ 
A)  $\theta = \frac { 3 \pi } { 2 } + 2 \mathrm { k } \pi$ 
B)  $\theta = \frac { \pi } { 2 } + 2 k \pi$ 
C)  $\theta = \pi + 2 k \pi$ 
D)  $\theta = 0 + 2 \mathrm { k } \pi$

Question 7

Solve the problem.
-If $\cos 2 \theta = - \frac { 24 } { 25 }$, and $\frac { \pi } { 2 } < 2 \theta < \pi$, then find $\sin \theta$.

Accepted Answer

A)  $\frac { 7 } { 5 }$ 
B)  $- \frac { 7 } { 5 }$ 
C)  $\frac { 7 \sqrt { 2 } } { 10 }$ 
D)  $- \frac { 7 \sqrt { 2 } } { 10 }$ 
A)  $\frac { 7 } { 5 }$ 
B)  $- \frac { 7 } { 5 }$ 
C)  $\frac { 7 \sqrt { 2 } } { 10 }$ 
D)  $- \frac { 7 \sqrt { 2 } } { 10 }$

Question 8

Choose the one alternative that best completes the statement or answers the question.
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 9

Find the exact value under the given conditions.
-$\sin \alpha = - \frac { 24 } { 25 } , \pi < \alpha < \frac { 3 \pi } { 2 } ; \quad \tan \beta = - \frac { 2 \sqrt { 21 } } { 21 } , \frac { \pi } { 2 } < \beta < \pi \quad$ Find $\cos ( \alpha + \beta )$.

Accepted Answer

A)  $\frac { 48 - 7 \sqrt { 21 } } { 125 }$ 
B)  $\frac { 48 + 7 \sqrt { 21 } } { 125 }$ 
C)  $\frac { 14 - 24 \sqrt { 21 } } { 125 }$ 
D)  $\frac { - 14 - 24 \sqrt { 21 } } { 125 }$ 
A)  $\frac { 48 - 7 \sqrt { 21 } } { 125 }$ 
B)  $\frac { 48 + 7 \sqrt { 21 } } { 125 }$ 
C)  $\frac { 14 - 24 \sqrt { 21 } } { 125 }$ 
D)  $\frac { - 14 - 24 \sqrt { 21 } } { 125 }$

Question 10

Choose the one alternative that best completes the statement or answers the question.
Complete the identity.
-$\sin ( 2 \theta ) \tan \theta + \cos ( 2 \theta ) = ?$

Accepted Answer

A)  $2 \cos ( 2 \theta )$ 
B)  $\cos ( 3 \theta )$ 
C)  1 
D)  $\sec ( 2 \theta )$ 
A)  $2 \cos ( 2 \theta )$ 
B)  $\cos ( 3 \theta )$ 
C)  1 
D)  $\sec ( 2 \theta )$

Question 11

Express the sum or difference as a product of sines and/or cosines.
-$\cos ( 6 \theta ) + \cos ( 4 \theta )$

Accepted Answer

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

Question 12

Choose the one alternative that best completes the statement or answers the question.
Express the product as a sum containing only sines or cosines.
-$\sin ( 4 \theta ) \cos ( 2 \theta )$

Accepted Answer

A)  $\sin \cos \left( 8 \theta ^ { 2 } \right)$ 
B)  $\frac { 1 } { 2 } [ \sin ( 6 \theta ) + \sin ( 2 \theta ) ]$ 
C)  $\frac { 1 } { 2 } [ \sin ( 6 \theta ) + \cos ( 2 \theta ) ]$ 
D)  $\frac { 1 } { 2 } [ \cos ( 6 \theta ) - \cos ( 2 \theta ) ]$ 
A)  $\sin \cos \left( 8 \theta ^ { 2 } \right)$ 
B)  $\frac { 1 } { 2 } [ \sin ( 6 \theta ) + \sin ( 2 \theta ) ]$ 
C)  $\frac { 1 } { 2 } [ \sin ( 6 \theta ) + \cos ( 2 \theta ) ]$ 
D)  $\frac { 1 } { 2 } [ \cos ( 6 \theta ) - \cos ( 2 \theta ) ]$

Question 13

Write the word or phrase that best completes each statement or answers the question.
-Wildlife management personnel use predator-prey equations to model the populations of certain predators and
their prey in the wild. Suppose the population M of a predator after t months is given by $$M = 750 + 125 \sin \frac { \pi } { 6 } t$$ while the population N of its primary prey is given by $$N = 12,250 + 3050 \cos \frac { \pi } { 6 } t$$ Find the values of $t , 0 \leq t < 12$, for which the predator population is 875 . Find the values of $t , 0 \leq t < 12$, for which the prey population is 10,725 .

Accepted Answer

The answer of Write the word or phrase that best...

Question 14

Find the exact value of the expression. Do not use a calculator.
-$$\sin ^ { - 1 } \left[ \sin \left( \frac { \pi } { 5 } \right) \right]$$

Accepted Answer

A)  $5 \pi$ 
B)  $\frac { \pi } { 5 }$ 
C)  $- 5 \pi$ 
D)  $- \frac { \pi } { 5 }$ 
A)  $5 \pi$ 
B)  $\frac { \pi } { 5 }$ 
C)  $- 5 \pi$ 
D)  $- \frac { \pi } { 5 }$

Question 15

Choose the one alternative that best completes the statement or answers the question.
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 16

Write the word or phrase that best completes each statement or answers the question.
-$$\sin 4 u = 2 \sin 2 u \cos 2 u$$

Accepted Answer

The answer of Write the word or phrase that best...

Question 17

Write the word or phrase that best completes each statement or answers the question.
-$$\sin ^ { 3 } 3 x = \frac { 1 } { 2 } ( \sin 3 x ) ( 1 - \cos 6 x )$$

Accepted Answer

The answer of Write the word or phrase that best...

Question 18

Find the exact value of the expression.
-$\sin \left( - \frac { 11 \pi } { 12 } \right)$

Accepted Answer

A)  $\frac { \sqrt { 2 } - \sqrt { 6 } } { 4 }$ 
B)  $\frac { \sqrt { 2 } + \sqrt { 6 } } { 4 }$ 
C)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
D)  $- \frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$ 
A)  $\frac { \sqrt { 2 } - \sqrt { 6 } } { 4 }$ 
B)  $\frac { \sqrt { 2 } + \sqrt { 6 } } { 4 }$ 
C)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
D)  $- \frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$

Question 19

Find the exact value of the expression.
-$\cos \left( \tan ^ { - 1 } \frac { 4 } { 3 } - \sin ^ { - 1 } \frac { 3 } { 5 } \right)$

Accepted Answer

A)  1 
B)  $\frac { 2 \sqrt { 6 } } { 5 }$ 
C)  $\frac { 2 \sqrt { 3 } } { 5 }$ 
D)  $\frac { 24 } { 25 }$ 
A)  1 
B)  $\frac { 2 \sqrt { 6 } } { 5 }$ 
C)  $\frac { 2 \sqrt { 3 } } { 5 }$ 
D)  $\frac { 24 } { 25 }$

Question 20

Express the sum or difference as a product of sines and/or cosines.
-$\sin ( 4 \theta ) - \sin ( 2 \theta )$

Accepted Answer

A)  $\sin ( 3 \theta ) \cos \theta$ 
B)  $\sin \theta \cos ( 3 \theta )$ 
C)  $2 \sin ( 3 \theta ) \cos \theta$ 
D)  $2 \sin \theta \cos ( 3 \theta )$ 
A)  $\sin ( 3 \theta ) \cos \theta$ 
B)  $\sin \theta \cos ( 3 \theta )$ 
C)  $2 \sin ( 3 \theta ) \cos \theta$ 
D)  $2 \sin \theta \cos ( 3 \theta )$

Express the sum or difference as a product of sines and/or cosines. - $\sin ( 6 \theta ) - \sin ( 4 \theta )$

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - $\cos \left( 2 \sin ^ { - 1 } \frac { 1 } { 4 } \right)$

Use the Half-angle Formulas to find the exact value of the trigonometric function. - $\cos 75 ^ { \circ }$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\sec \theta - \frac { 1 } { \sec \theta } =$ ?

Solve the equation. Give a general formula for all the solutions. - $\cos \theta = 1$

Solve the problem. -If $\cos 2 \theta = - \frac { 24 } { 25 }$ , and $\frac { \pi } { 2 } < 2 \theta < \pi$ , then find $\sin \theta$ .

Choose the one alternative that best completes the statement or answers the question. Express the product as a sum containing only sines or cosines. - $- 2 \sin ( 5 \theta ) \sin \theta$

Find the exact value under the given conditions. - $\sin \alpha = - \frac { 24 } { 25 } , \pi < \alpha < \frac { 3 \pi } { 2 } ; \quad \tan \beta = - \frac { 2 \sqrt { 21 } } { 21 } , \frac { \pi } { 2 } < \beta < \pi \quad$ Find $\cos ( \alpha + \beta )$ .

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\sin ( 2 \theta ) \tan \theta + \cos ( 2 \theta ) = ?$

Express the sum or difference as a product of sines and/or cosines. - $\cos ( 6 \theta ) + \cos ( 4 \theta )$

Choose the one alternative that best completes the statement or answers the question. Express the product as a sum containing only sines or cosines. - $\sin ( 4 \theta ) \cos ( 2 \theta )$

Find the exact value of the expression. Do not use a calculator. - $\sin ^ { - 1 } \left[ \sin \left( \frac { \pi } { 5 } \right) \right]$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\cos ( 4 \theta ) = \text { ? }$

Write the word or phrase that best completes each statement or answers the question. - $\sin 4 u = 2 \sin 2 u \cos 2 u$

Write the word or phrase that best completes each statement or answers the question. - $\sin ^ { 3 } 3 x = \frac { 1 } { 2 } ( \sin 3 x ) ( 1 - \cos 6 x )$

Find the exact value of the expression. - $\sin \left( - \frac { 11 \pi } { 12 } \right)$

Find the exact value of the expression. - $\cos \left( \tan ^ { - 1 } \frac { 4 } { 3 } - \sin ^ { - 1 } \frac { 3 } { 5 } \right)$

Express the sum or difference as a product of sines and/or cosines. - $\sin ( 4 \theta ) - \sin ( 2 \theta )$

Functions and Their Graphs

Linear and Quadratic Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Applications of Trigonometric Functions

Polar Coordinates; Vectors

Analytic Geometry

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Foundations: a Prelude to Functions

Graphing Utilities

Filters

Exam 6: Analytic Trigonometry

Express the sum or difference as a product of sines and/or cosines. - sin⁡(6θ)−sin⁡(4θ)\sin ( 6 \theta ) - \sin ( 4 \theta )sin(6θ)−sin(4θ)

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - cos⁡(2sin⁡−114)\cos \left( 2 \sin ^ { - 1 } \frac { 1 } { 4 } \right)cos(2sin−141​)

Use the Half-angle Formulas to find the exact value of the trigonometric function. - cos⁡75∘\cos 75 ^ { \circ }cos75∘

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - sec⁡θ−1sec⁡θ=\sec \theta - \frac { 1 } { \sec \theta } =secθ−secθ1​= ?

Solve the equation. Give a general formula for all the solutions. - cos⁡θ=1\cos \theta = 1cosθ=1

Solve the problem. -If cos⁡2θ=−2425\cos 2 \theta = - \frac { 24 } { 25 }cos2θ=−2524​ , and π2<2θ<π\frac { \pi } { 2 } < 2 \theta < \pi2π​<2θ<π , then find sin⁡θ\sin \thetasinθ .

Choose the one alternative that best completes the statement or answers the question. Express the product as a sum containing only sines or cosines. - −2sin⁡(5θ)sin⁡θ- 2 \sin ( 5 \theta ) \sin \theta−2sin(5θ)sinθ

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - sin⁡(2θ)tan⁡θ+cos⁡(2θ)=?\sin ( 2 \theta ) \tan \theta + \cos ( 2 \theta ) = ?sin(2θ)tanθ+cos(2θ)=?

Express the sum or difference as a product of sines and/or cosines. - cos⁡(6θ)+cos⁡(4θ)\cos ( 6 \theta ) + \cos ( 4 \theta )cos(6θ)+cos(4θ)

Choose the one alternative that best completes the statement or answers the question. Express the product as a sum containing only sines or cosines. - sin⁡(4θ)cos⁡(2θ)\sin ( 4 \theta ) \cos ( 2 \theta )sin(4θ)cos(2θ)

Find the exact value of the expression. Do not use a calculator. - sin⁡−1[sin⁡(π5)]\sin ^ { - 1 } \left[ \sin \left( \frac { \pi } { 5 } \right) \right]sin−1[sin(5π​)]

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - cos⁡(4θ)= ? \cos ( 4 \theta ) = \text { ? }cos(4θ)= ?

Write the word or phrase that best completes each statement or answers the question. - sin⁡4u=2sin⁡2ucos⁡2u\sin 4 u = 2 \sin 2 u \cos 2 usin4u=2sin2ucos2u

Write the word or phrase that best completes each statement or answers the question. - sin⁡33x=12(sin⁡3x)(1−cos⁡6x)\sin ^ { 3 } 3 x = \frac { 1 } { 2 } ( \sin 3 x ) ( 1 - \cos 6 x )sin33x=21​(sin3x)(1−cos6x)

Find the exact value of the expression. - sin⁡(−11π12)\sin \left( - \frac { 11 \pi } { 12 } \right)sin(−1211π​)

Find the exact value of the expression. - cos⁡(tan⁡−143−sin⁡−135)\cos \left( \tan ^ { - 1 } \frac { 4 } { 3 } - \sin ^ { - 1 } \frac { 3 } { 5 } \right)cos(tan−134​−sin−153​)

Express the sum or difference as a product of sines and/or cosines. - sin⁡(4θ)−sin⁡(2θ)\sin ( 4 \theta ) - \sin ( 2 \theta )sin(4θ)−sin(2θ)

Functions and Their Graphs

Linear and Quadratic Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Applications of Trigonometric Functions

Polar Coordinates; Vectors

Analytic Geometry

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Foundations: a Prelude to Functions

Graphing Utilities

Filters

Express the sum or difference as a product of sines and/or cosines. - $\sin ( 6 \theta ) - \sin ( 4 \theta )$

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - $\cos \left( 2 \sin ^ { - 1 } \frac { 1 } { 4 } \right)$

Use the Half-angle Formulas to find the exact value of the trigonometric function. - $\cos 75 ^ { \circ }$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\sec \theta - \frac { 1 } { \sec \theta } =$ ?

Solve the equation. Give a general formula for all the solutions. - $\cos \theta = 1$

Solve the problem. -If $\cos 2 \theta = - \frac { 24 } { 25 }$ , and $\frac { \pi } { 2 } < 2 \theta < \pi$ , then find $\sin \theta$ .

Choose the one alternative that best completes the statement or answers the question. Express the product as a sum containing only sines or cosines. - $- 2 \sin ( 5 \theta ) \sin \theta$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\sin ( 2 \theta ) \tan \theta + \cos ( 2 \theta ) = ?$

Express the sum or difference as a product of sines and/or cosines. - $\cos ( 6 \theta ) + \cos ( 4 \theta )$

Choose the one alternative that best completes the statement or answers the question. Express the product as a sum containing only sines or cosines. - $\sin ( 4 \theta ) \cos ( 2 \theta )$

Find the exact value of the expression. Do not use a calculator. - $\sin ^ { - 1 } \left[ \sin \left( \frac { \pi } { 5 } \right) \right]$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\cos ( 4 \theta ) = \text { ? }$

Write the word or phrase that best completes each statement or answers the question. - $\sin 4 u = 2 \sin 2 u \cos 2 u$

Write the word or phrase that best completes each statement or answers the question. - $\sin ^ { 3 } 3 x = \frac { 1 } { 2 } ( \sin 3 x ) ( 1 - \cos 6 x )$

Find the exact value of the expression. - $\sin \left( - \frac { 11 \pi } { 12 } \right)$

Find the exact value of the expression. - $\cos \left( \tan ^ { - 1 } \frac { 4 } { 3 } - \sin ^ { - 1 } \frac { 3 } { 5 } \right)$

Express the sum or difference as a product of sines and/or cosines. - $\sin ( 4 \theta ) - \sin ( 2 \theta )$