Question 1

Express the sum or difference as a product of sines and/or cosines.
-$$\cos \frac { 5 \theta } { 2 } + \cos \frac { 3 \theta } { 2 }$$

Accepted Answer

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

Question 2

Establish the identity.
-$$\sin ( x + y ) - \sin ( x - y ) = 2 \cos x \sin y$$

Accepted Answer

The answer of Establish the identity.
-\[\sin ( x + y...

Question 3

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function.
-$\cos \theta = \frac { 1 } { 4 } , \csc \theta > 0 \quad$ Find $\sin \frac { \theta } { 2 }$.

Accepted Answer

A)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
B)  $\frac { \sqrt { 6 } } { 4 }$ 
C)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
D)  $\frac { \sqrt { 10 } } { 4 }$ 
A)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
B)  $\frac { \sqrt { 6 } } { 4 }$ 
C)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
D)  $\frac { \sqrt { 10 } } { 4 }$

Question 4

Establish the identity.
-$$\cos \left( x + \frac { \pi } { 6 } \right) = \frac { \sqrt { 3 } } { 2 } \cos x - \frac { 1 } { 2 } \sin x$$

Accepted Answer

The answer of Establish the identity.
-\[\cos \left( x + \frac...

Question 5

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

Accepted Answer

The answer of Establish the identity.
-\[\frac { 1 + \csc...

Question 6

Establish the identity.
-$$\cot ^ { 3 } x = \cot x \left( \csc ^ { 2 } x - 1 \right)$$

Accepted Answer

The answer of Establish the identity.
-\[\cot ^ { 3 }...

Question 7

Use a calculator to find the value of the expression rounded to two decimal places.
-$\sin ^ { - 1 } \left( \frac { \sqrt { 5 } } { 5 } \right)$

Accepted Answer

A)  63.44 
B)  1.11 
C)  26.57 
D)  0.46 
A)  63.44 
B)  1.11 
C)  26.57 
D)  0.46

Question 8

Express the product as a sum containing only sines or cosines.
-$\cos ( 5 \theta ) \cos ( 4 \theta )$

Accepted Answer

A)  $\frac { 1 } { 2 } [ \cos ( 9 \theta ) - \cos \theta ]$ 
B)  $\frac { 1 } { 2 } [ \cos \theta + \cos ( 9 \theta ) ]$ 
C)  $\cos ^ { 2 } \left( 20 \theta ^ { 2 } \right)$ 
D)  $\frac { 1 } { 2 } [ \cos ( 9 \theta ) - \sin \theta ]$ 
A)  $\frac { 1 } { 2 } [ \cos ( 9 \theta ) - \cos \theta ]$ 
B)  $\frac { 1 } { 2 } [ \cos \theta + \cos ( 9 \theta ) ]$ 
C)  $\cos ^ { 2 } \left( 20 \theta ^ { 2 } \right)$ 
D)  $\frac { 1 } { 2 } [ \cos ( 9 \theta ) - \sin \theta ]$

Question 9

Use the information given about the angle $$\theta , 0 \leq \theta \leq 2 \pi$$ , to find the exact value of the indicated trigonometric function.
-$\sin \theta = \frac { 2 \sqrt { 10 } } { 7 } , \quad \tan \theta < 0 \quad$ Find $\sin ( 2 \theta )$

Accepted Answer

A)  $\frac { - 12 \sqrt { 10 } } { 49 }$ 
B)  $- \frac { 31 } { 49 }$ 
C)  $\frac { 12 \sqrt { 10 } } { 49 }$ 
D)  $\frac { 31 } { 49 }$ 
A)  $\frac { - 12 \sqrt { 10 } } { 49 }$ 
B)  $- \frac { 31 } { 49 }$ 
C)  $\frac { 12 \sqrt { 10 } } { 49 }$ 
D)  $\frac { 31 } { 49 }$

Question 10

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

Accepted Answer

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

Question 11

Find the exact solution of the equation.
-$$\sin ^ { - 1 } x = \frac { \pi } { 2 }$$

Accepted Answer

A)  $x = 1$ 
B)  $x = \pi$ 
C)  $x = - 1$ 
D)  $x = 0$ 
A)  $x = 1$ 
B)  $x = \pi$ 
C)  $x = - 1$ 
D)  $x = 0$

Question 12

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

Accepted Answer

A)  $v \sqrt { 1 - u ^ { 2 } } - u \sqrt { 1 - v ^ { 2 } }$ 
B)  $u v + \left( \sqrt { 1 - u ^ { 2 } } \right) \left( \sqrt { 1 - v ^ { 2 } } \right)$ 
C)  $u v - \left( \sqrt { 1 - u ^ { 2 } } \right) \left( \sqrt { 1 - v ^ { 2 } } \right)$ 
D)  $v \sqrt { 1 - u ^ { 2 } } + u \sqrt { 1 - v ^ { 2 } }$ 
A)  $v \sqrt { 1 - u ^ { 2 } } - u \sqrt { 1 - v ^ { 2 } }$ 
B)  $u v + \left( \sqrt { 1 - u ^ { 2 } } \right) \left( \sqrt { 1 - v ^ { 2 } } \right)$ 
C)  $u v - \left( \sqrt { 1 - u ^ { 2 } } \right) \left( \sqrt { 1 - v ^ { 2 } } \right)$ 
D)  $v \sqrt { 1 - u ^ { 2 } } + u \sqrt { 1 - v ^ { 2 } }$

Question 13

Solve the equation on the interval $$0 \leq \theta < 2 \pi$$
-$\sec ^ { 2 } \theta - 2 = \tan ^ { 2 } \theta$

Accepted Answer

A)  $\frac { \pi } { 6 }$ 
B)  $\frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 4 }$ 
D)  No solution 
A)  $\frac { \pi } { 6 }$ 
B)  $\frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 4 }$ 
D)  No solution

Question 14

Complete the identity.
-$$\frac { \sin ( \alpha - \beta ) } { \sin ( \alpha + \beta ) } = ?$$

Accepted Answer

A)  $\frac { \tan \alpha \tan \beta - 1 } { \tan \alpha \tan \beta + 1 }$ 
B)  $\frac { \tan \alpha - \tan \beta } { \tan \alpha + \tan \beta }$ 
C)  -1 
D)  $\frac { \sin \alpha - \sin \beta } { \sin \alpha + \sin \beta }$ 
A)  $\frac { \tan \alpha \tan \beta - 1 } { \tan \alpha \tan \beta + 1 }$ 
B)  $\frac { \tan \alpha - \tan \beta } { \tan \alpha + \tan \beta }$ 
C)  -1 
D)  $\frac { \sin \alpha - \sin \beta } { \sin \alpha + \sin \beta }$

Question 15

Find the inverse function $\mathrm { f } ^ { - 1 }$ of the function f.
-$$f ( x ) = \cos ( x - 9 ) - 5$$

Accepted Answer

A)  $f ^ { - 1 } ( x ) = \cos ^ { - 1 } ( x - 5 ) - 9$ 
B)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \cos ^ { - 1 } ( \mathrm { x } + 5 ) + 9$ 
C)  $f ^ { - 1 } ( x ) = \cos ^ { - 1 } ( x + 9 ) + 5$ 
D)  $f ^ { - 1 } ( x ) = \cos ^ { - 1 } ( x - 9 ) - 5$ 
A)  $f ^ { - 1 } ( x ) = \cos ^ { - 1 } ( x - 5 ) - 9$ 
B)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \cos ^ { - 1 } ( \mathrm { x } + 5 ) + 9$ 
C)  $f ^ { - 1 } ( x ) = \cos ^ { - 1 } ( x + 9 ) + 5$ 
D)  $f ^ { - 1 } ( x ) = \cos ^ { - 1 } ( x - 9 ) - 5$

Question 16

Use a calculator to find the value of the expression rounded to two decimal places.
-$\sin ^ { - 1 } ( 0.7 )$

Accepted Answer

A)  44.43 
B)  0.78 
C)  45.57 
D)  0.80 
A)  44.43 
B)  0.78 
C)  45.57 
D)  0.80

Question 17

Solve the equation on the interval [0, 2π).
-Suppose $f ( x ) = 6 \csc \theta - 1$. Solve $f ( x ) = 5$.

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

Complete the identity.
-$\tan ^ { 2 } \theta - 3 \sin \theta \tan \theta \sec \theta = ?$

Accepted Answer

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

Question 20

Find the exact value of the expression.
-$\cos \left( \frac { 7 \pi } { 12 } \right) \cos \left( \frac { 5 \pi } { 12 } \right) + \sin \left( \frac { 7 \pi } { 12 } \right) \sin \left( \frac { 5 \pi } { 12 } \right)$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  1 
D)  $\frac { \sqrt { 3 } } { 2 }$ 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  1 
D)  $\frac { \sqrt { 3 } } { 2 }$

Express the sum or difference as a product of sines and/or cosines. - $\cos \frac { 5 \theta } { 2 } + \cos \frac { 3 \theta } { 2 }$

Establish the identity. - $\sin ( x + y ) - \sin ( x - y ) = 2 \cos x \sin y$

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function. - $\cos \theta = \frac { 1 } { 4 } , \csc \theta > 0 \quad$ Find $\sin \frac { \theta } { 2 }$ .

Establish the identity. - $\cos \left( x + \frac { \pi } { 6 } \right) = \frac { \sqrt { 3 } } { 2 } \cos x - \frac { 1 } { 2 } \sin x$

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

Establish the identity. - $\cot ^ { 3 } x = \cot x \left( \csc ^ { 2 } x - 1 \right)$

Use a calculator to find the value of the expression rounded to two decimal places. - $\sin ^ { - 1 } \left( \frac { \sqrt { 5 } } { 5 } \right)$

Express the product as a sum containing only sines or cosines. - $\cos ( 5 \theta ) \cos ( 4 \theta )$

Use the information given about the angle $\theta , 0 \leq \theta \leq 2 \pi$ , to find the exact value of the indicated trigonometric function. - $\sin \theta = \frac { 2 \sqrt { 10 } } { 7 } , \quad \tan \theta < 0 \quad$ Find $\sin ( 2 \theta )$

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

Find the exact solution of the equation. - $\sin ^ { - 1 } x = \frac { \pi } { 2 }$

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

Solve the equation on the interval $0 \leq \theta < 2 \pi$ - $\sec ^ { 2 } \theta - 2 = \tan ^ { 2 } \theta$

Complete the identity. - $\frac { \sin ( \alpha - \beta ) } { \sin ( \alpha + \beta ) } = ?$

Find the inverse function $\mathrm { f } ^ { - 1 }$ of the function f. - $f ( x ) = \cos ( x - 9 ) - 5$

Use a calculator to find the value of the expression rounded to two decimal places. - $\sin ^ { - 1 } ( 0.7 )$

Solve the equation on the interval [0, 2π). -Suppose $f ( x ) = 6 \csc \theta - 1$ . Solve $f ( x ) = 5$ .

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

Complete the identity. - $\tan ^ { 2 } \theta - 3 \sin \theta \tan \theta \sec \theta = ?$

Find the exact value of the expression. - $\cos \left( \frac { 7 \pi } { 12 } \right) \cos \left( \frac { 5 \pi } { 12 } \right) + \sin \left( \frac { 7 \pi } { 12 } \right) \sin \left( \frac { 5 \pi } { 12 } \right)$

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

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Exam 6: Analytic Trigonometry

Express the sum or difference as a product of sines and/or cosines. - cos⁡5θ2+cos⁡3θ2\cos \frac { 5 \theta } { 2 } + \cos \frac { 3 \theta } { 2 }cos25θ​+cos23θ​

Establish the identity. - sin⁡(x+y)−sin⁡(x−y)=2cos⁡xsin⁡y\sin ( x + y ) - \sin ( x - y ) = 2 \cos x \sin ysin(x+y)−sin(x−y)=2cosxsiny

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function. - cos⁡θ=14,csc⁡θ>0\cos \theta = \frac { 1 } { 4 } , \csc \theta > 0 \quadcosθ=41​,cscθ>0 Find sin⁡θ2\sin \frac { \theta } { 2 }sin2θ​ .

Establish the identity. - cos⁡(x+π6)=32cos⁡x−12sin⁡x\cos \left( x + \frac { \pi } { 6 } \right) = \frac { \sqrt { 3 } } { 2 } \cos x - \frac { 1 } { 2 } \sin xcos(x+6π​)=23​​cosx−21​sinx

Establish the identity. - 1+csc⁡xsec⁡x=cos⁡x+cot⁡x\frac { 1 + \csc x } { \sec x } = \cos x + \cot xsecx1+cscx​=cosx+cotx

Establish the identity. - cot⁡3x=cot⁡x(csc⁡2x−1)\cot ^ { 3 } x = \cot x \left( \csc ^ { 2 } x - 1 \right)cot3x=cotx(csc2x−1)

Use a calculator to find the value of the expression rounded to two decimal places. - sin⁡−1(55)\sin ^ { - 1 } \left( \frac { \sqrt { 5 } } { 5 } \right)sin−1(55​​)

Express the product as a sum containing only sines or cosines. - cos⁡(5θ)cos⁡(4θ)\cos ( 5 \theta ) \cos ( 4 \theta )cos(5θ)cos(4θ)

Find the exact value of the expression. - tan⁡(cos⁡−113)\tan \left( \cos ^ { - 1 } \frac { 1 } { 3 } \right)tan(cos−131​)

Find the exact solution of the equation. - sin⁡−1x=π2\sin ^ { - 1 } x = \frac { \pi } { 2 }sin−1x=2π​

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

Solve the equation on the interval 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π - sec⁡2θ−2=tan⁡2θ\sec ^ { 2 } \theta - 2 = \tan ^ { 2 } \thetasec2θ−2=tan2θ

Complete the identity. - sin⁡(α−β)sin⁡(α+β)=?\frac { \sin ( \alpha - \beta ) } { \sin ( \alpha + \beta ) } = ?sin(α+β)sin(α−β)​=?

Find the inverse function f−1\mathrm { f } ^ { - 1 }f−1 of the function f. - f(x)=cos⁡(x−9)−5f ( x ) = \cos ( x - 9 ) - 5f(x)=cos(x−9)−5

Use a calculator to find the value of the expression rounded to two decimal places. - sin⁡−1(0.7)\sin ^ { - 1 } ( 0.7 )sin−1(0.7)

Solve the equation on the interval [0, 2π). -Suppose f(x)=6csc⁡θ−1f ( x ) = 6 \csc \theta - 1f(x)=6cscθ−1 . Solve f(x)=5f ( x ) = 5f(x)=5 .

Use the Half-angle Formulas to find the exact value of the trigonometric function. - tan⁡75∘\tan 75 ^ { \circ }tan75∘

Complete the identity. - tan⁡2θ−3sin⁡θtan⁡θsec⁡θ=?\tan ^ { 2 } \theta - 3 \sin \theta \tan \theta \sec \theta = ?tan2θ−3sinθtanθsecθ=?