Question 1

Use the figure to find the exact value of the trigonometric function.
-$\tan \theta = \frac { 8 } { 15 } , \theta$ lies in quadrant III $\quad$ Find $\sin 2 \theta$

Accepted Answer

A)  $\frac { 240 } { 289 }$ 
B)  $- \frac { 161 } { 289 }$ 
C)  $\frac { 161 } { 289 }$ 
D)  $- \frac { 240 } { 289 }$ 
A)  $\frac { 240 } { 289 }$ 
B)  $- \frac { 161 } { 289 }$ 
C)  $\frac { 161 } { 289 }$ 
D)  $- \frac { 240 } { 289 }$

Question 2

Use the graph to complete the identity.
-$\frac { \cos x \tan x - 4 \tan x + 5 \cos x - 20 } { \tan x + 5 } =$ ?

Accepted Answer

A)  $\cos x - 4$ 
B)  $\cos x + 4$ 
C)  $\sin x - 5$ 
D)  $\sin x + 5 \cos x$ 
A)  $\cos x - 4$ 
B)  $\cos x + 4$ 
C)  $\sin x - 5$ 
D)  $\sin x + 5 \cos x$

Question 3

Solve the equation on the interval [0, 2π).
-$\cos x + 2 \cos x \sin x = 0$

Accepted Answer

A)  $\frac { \pi } { 2 } , \frac { 7 \pi } { 6 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 6 }$ 
B)  $0 , \frac { \pi } { 2 } , \frac { 7 \pi } { 6 } , \frac { 3 \pi } { 2 }$ 
C)  $\frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
D)  $\frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 } , 2 \pi$ 
A)  $\frac { \pi } { 2 } , \frac { 7 \pi } { 6 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 6 }$ 
B)  $0 , \frac { \pi } { 2 } , \frac { 7 \pi } { 6 } , \frac { 3 \pi } { 2 }$ 
C)  $\frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
D)  $\frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 } , 2 \pi$

Question 4

Express the sum or difference as a product.
-$$\cos \frac { 7 x } { 2 } + \cos \frac { 5 x } { 2 }$$

Accepted Answer

A)  $2 \cos 3 x \cos \frac { x } { 2 }$ 
B)  $2 \sin 3 x \sin x$ 
C)  $2 \sin 3 x \sin \frac { x } { 2 }$ 
D)  $2 \cos 3 x$ 
A)  $2 \cos 3 x \cos \frac { x } { 2 }$ 
B)  $2 \sin 3 x \sin x$ 
C)  $2 \sin 3 x \sin \frac { x } { 2 }$ 
D)  $2 \cos 3 x$

Question 5

Express the product as a sum or difference.
-$\sin 6 x \sin 2 x$

Accepted Answer

A)  $\frac { 1 } { 2 } ( \cos 4 x - \cos 8 x )$ 
B)  $\sin ^ { 2 } 12 x ^ { 2 }$ 
C)  $\frac { 1 } { 2 } ( \cos 8 x - \cos 4 x )$ 
D)  $\frac { 1 } { 2 } ( \sin 8 x + \cos 4 x )$ 
A)  $\frac { 1 } { 2 } ( \cos 4 x - \cos 8 x )$ 
B)  $\sin ^ { 2 } 12 x ^ { 2 }$ 
C)  $\frac { 1 } { 2 } ( \cos 8 x - \cos 4 x )$ 
D)  $\frac { 1 } { 2 } ( \sin 8 x + \cos 4 x )$

Question 6

Use Sum and Difference Formulas for Tangents
Find the exact value by using a difference identity.
-$\tan 255 ^ { \circ }$

Accepted Answer

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

Question 7

Use the given information to find the exact value of the trigonometric function.
-$\csc \theta = - \frac { 6 } { 5 } , \tan \theta > 0 \quad$ Find $\cos \frac { \theta } { 2 }$.

Accepted Answer

A)  $- \frac { \sqrt { 18 - 3 \sqrt { 11 } } } { 6 }$ 
B)  $\frac { \sqrt { 18 + 3 \sqrt { 11 } } } { 6 }$ 
C)  $\frac { \sqrt { 11 } } { 12 }$ 
D)  $- \frac { \sqrt { 6 + \sqrt { 11 } } } { 12 }$ 
A)  $- \frac { \sqrt { 18 - 3 \sqrt { 11 } } } { 6 }$ 
B)  $\frac { \sqrt { 18 + 3 \sqrt { 11 } } } { 6 }$ 
C)  $\frac { \sqrt { 11 } } { 12 }$ 
D)  $- \frac { \sqrt { 6 + \sqrt { 11 } } } { 12 }$

Question 8

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

Accepted Answer

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

Question 9

Use Identities to Solve Trigonometric Equations
Solve the equation on the interval [0, 2π).
-$\sin 2 x + \sin x = 0$

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 10

Use Identities to Solve Trigonometric Equations
Solve the equation on the interval [0, 2π).
-$$\cos \left( x + \frac { \pi } { 3 } \right) + \cos \left( x - \frac { \pi } { 3 } \right) = 1$$

Accepted Answer

A)  0 
B)  $\frac { \pi } { 2 } , \frac { \pi } { 6 }$ 
C)  $\frac { \pi } { 6 } , \frac { \pi } { 4 }$ 
D)  $\frac { \pi } { 2 } , \frac { \pi } { 4 }$ 
A)  0 
B)  $\frac { \pi } { 2 } , \frac { \pi } { 6 }$ 
C)  $\frac { \pi } { 6 } , \frac { \pi } { 4 }$ 
D)  $\frac { \pi } { 2 } , \frac { \pi } { 4 }$

Question 11

Use the Formula for the Cosine of the Difference of Two Angles
-$\cos \left( \frac { 5 \pi } { 12 } \right) \cos \left( \frac { \pi } { 4 } \right) + \sin \left( \frac { 5 \pi } { 12 } \right) \sin \left( \frac { \pi } { 4 } \right)$

Accepted Answer

A)  $\cos \left( \frac { \pi } { 6 } \right)$ 
B)  $\cos \left( \frac { \pi } { 3 } \right)$ 
C)  $\cos \left( \frac { 2 \pi } { 3 } \right)$ 
D)  $\cos \left( \frac { 5 \pi } { 6 } \right)$ Find the exact value of the expression. 
A)  $\cos \left( \frac { \pi } { 6 } \right)$ 
B)  $\cos \left( \frac { \pi } { 3 } \right)$ 
C)  $\cos \left( \frac { 2 \pi } { 3 } \right)$ 
D)  $\cos \left( \frac { 5 \pi } { 6 } \right)$ Find the exact value of the expression.

Question 12

Use the given information to find the exact value of the trigonometric function.
-$\sin \theta = - \frac { 3 } { 5 } , \quad \theta$ lies in quadrant IV Find $\sin \frac { \theta } { 2 }$.

Accepted Answer

A)  $\frac { \sqrt { 10 } } { 10 }$ 
B)  $\frac { \sqrt { 5 } } { 5 }$ 
C)  $- \frac { \sqrt { 5 } } { 5 }$ 
D)  $- \frac { \sqrt { 30 } } { 10 }$ 
A)  $\frac { \sqrt { 10 } } { 10 }$ 
B)  $\frac { \sqrt { 5 } } { 5 }$ 
C)  $- \frac { \sqrt { 5 } } { 5 }$ 
D)  $- \frac { \sqrt { 30 } } { 10 }$

Question 13

Write the expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression.
-$2 \sin 75 ^ { \circ } \cos 75 ^ { \circ }$

Accepted Answer

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

Question 14

Express the sum or difference as a product.
-$$\sin \frac { 9 x } { 2 } + \sin \frac { 5 x } { 2 }$$

Accepted Answer

A)  $2 \sin \frac { 7 x } { 2 } \cos x$ 
B)  $2 \sin \frac { 7 x } { 2 } \sin x$ 
C)  $2 \cos 7 x \sin x$ 
D)  $2 \sin 7 x$ 
A)  $2 \sin \frac { 7 x } { 2 } \cos x$ 
B)  $2 \sin \frac { 7 x } { 2 } \sin x$ 
C)  $2 \cos 7 x \sin x$ 
D)  $2 \sin 7 x$

Question 15

Use the given information to find the exact value of the expression.
-$\sin \alpha = \frac { 8 } { 17 } , \alpha$ lies in quadrant II, and $\cos \beta = \frac { 12 } { 13 } , \beta$ lies in quadrant I $\quad$ Find $\sin ( \alpha - \beta )$.

Accepted Answer

A)  $\frac { 171 } { 221 }$ 
B)  $\frac { 220 } { 221 }$ 
C)  $\frac { 140 } { 221 }$ 
D)  $\frac { 21 } { 221 }$ 
A)  $\frac { 171 } { 221 }$ 
B)  $\frac { 220 } { 221 }$ 
C)  $\frac { 140 } { 221 }$ 
D)  $\frac { 21 } { 221 }$

Question 16

Use Identities to Solve Trigonometric Equations
Solve the equation on the interval [0, 2π).
-$$\sec \frac { x } { 2 } = \cos \frac { x } { 2 }$$

Accepted Answer

A)  0 
B)  $\frac { \pi } { 4 } , \frac { 5 \pi } { 4 }$ 
C)  $\frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 }$ 
D)  $0 , \frac { \pi } { 4 } , \pi , \frac { 5 \pi } { 3 }$ 
A)  0 
B)  $\frac { \pi } { 4 } , \frac { 5 \pi } { 4 }$ 
C)  $\frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 }$ 
D)  $0 , \frac { \pi } { 4 } , \pi , \frac { 5 \pi } { 3 }$

Question 17

Use Sum and Difference Formulas for Tangents
Find the exact value by using a difference identity.
-$\tan 285 ^ { \circ }$

Accepted Answer

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

Question 18

Complete the identity.
-$\sin ^ { 2 } x + \tan ^ { 2 } x + \cos ^ { 2 } x = ?$

Accepted Answer

A)  $\sec ^ { 2 } x$ 
B)  $\tan ^ { 2 } x$ 
C)  $\cot ^ { 3 } x$ 
D)  $\sin x$ 
A)  $\sec ^ { 2 } x$ 
B)  $\tan ^ { 2 } x$ 
C)  $\cot ^ { 3 } x$ 
D)  $\sin x$

Question 19

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

Accepted Answer

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

Question 20

Use the Power-Reducing Formulas
-$3 \sin ^ { 2 } x \cos ^ { 2 } x$

Accepted Answer

A)  $\frac { 3 } { 8 } - \frac { 3 } { 8 } \cos 4 x$ 
B)  $\frac { 3 } { 4 } - \frac { 3 } { 4 } \cos 4 x$ 
C)  $\frac { 3 } { 8 } - \frac { 3 } { 8 } \cos 2 x$ 
D)  $\frac { 3 } { 4 } \sin 2 x$ 
A)  $\frac { 3 } { 8 } - \frac { 3 } { 8 } \cos 4 x$ 
B)  $\frac { 3 } { 4 } - \frac { 3 } { 4 } \cos 4 x$ 
C)  $\frac { 3 } { 8 } - \frac { 3 } { 8 } \cos 2 x$ 
D)  $\frac { 3 } { 4 } \sin 2 x$

Use the figure to find the exact value of the trigonometric function. - $\tan \theta = \frac { 8 } { 15 } , \theta$ lies in quadrant III $\quad$ Find $\sin 2 \theta$

Use the graph to complete the identity. - $\frac { \cos x \tan x - 4 \tan x + 5 \cos x - 20 } { \tan x + 5 } =$ ? $Use the graph to complete the identity. - \frac { \cos x \tan x - 4 \tan x + 5 \cos x - 20 } { \tan x + 5 } = ?$

Solve the equation on the interval [0, 2π). - $\cos x + 2 \cos x \sin x = 0$

Express the sum or difference as a product. - $\cos \frac { 7 x } { 2 } + \cos \frac { 5 x } { 2 }$

Express the product as a sum or difference. - $\sin 6 x \sin 2 x$

Use Sum and Difference Formulas for Tangents Find the exact value by using a difference identity. - $\tan 255 ^ { \circ }$

Use the given information to find the exact value of the trigonometric function. - $\csc \theta = - \frac { 6 } { 5 } , \tan \theta > 0 \quad$ Find $\cos \frac { \theta } { 2 }$ .

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

Use Identities to Solve Trigonometric Equations Solve the equation on the interval [0, 2π). - $\sin 2 x + \sin x = 0$

Use Identities to Solve Trigonometric Equations Solve the equation on the interval [0, 2π). - $\cos \left( x + \frac { \pi } { 3 } \right) + \cos \left( x - \frac { \pi } { 3 } \right) = 1$

Use the Formula for the Cosine of the Difference of Two Angles - $\cos \left( \frac { 5 \pi } { 12 } \right) \cos \left( \frac { \pi } { 4 } \right) + \sin \left( \frac { 5 \pi } { 12 } \right) \sin \left( \frac { \pi } { 4 } \right)$

Use the given information to find the exact value of the trigonometric function. - $\sin \theta = - \frac { 3 } { 5 } , \quad \theta$ lies in quadrant IV Find $\sin \frac { \theta } { 2 }$ .

Write the expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. - $2 \sin 75 ^ { \circ } \cos 75 ^ { \circ }$

Express the sum or difference as a product. - $\sin \frac { 9 x } { 2 } + \sin \frac { 5 x } { 2 }$

Use the given information to find the exact value of the expression. - $\sin \alpha = \frac { 8 } { 17 } , \alpha$ lies in quadrant II, and $\cos \beta = \frac { 12 } { 13 } , \beta$ lies in quadrant I $\quad$ Find $\sin ( \alpha - \beta )$ .

Use Identities to Solve Trigonometric Equations Solve the equation on the interval [0, 2π). - $\sec \frac { x } { 2 } = \cos \frac { x } { 2 }$

Use Sum and Difference Formulas for Tangents Find the exact value by using a difference identity. - $\tan 285 ^ { \circ }$

Complete the identity. - $\sin ^ { 2 } x + \tan ^ { 2 } x + \cos ^ { 2 } x = ?$

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

Use the Power-Reducing Formulas - $3 \sin ^ { 2 } x \cos ^ { 2 } x$

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Additional Topics in Trigonometry

Systems of Equations and Inequalities

Matrices and Determinants

Conic Sections and Analytic Geometry

Sequences, Induction, and Probability

Prerequisites: Fundamental Concepts of Algebra

Filters

Exam 6: Analytic Trigonometry

Use the figure to find the exact value of the trigonometric function. - tan⁡θ=815,θ\tan \theta = \frac { 8 } { 15 } , \thetatanθ=158​,θ lies in quadrant III \quad Find sin⁡2θ\sin 2 \thetasin2θ

Use the graph to complete the identity. - cos⁡xtan⁡x−4tan⁡x+5cos⁡x−20tan⁡x+5=\frac { \cos x \tan x - 4 \tan x + 5 \cos x - 20 } { \tan x + 5 } =tanx+5cosxtanx−4tanx+5cosx−20​= ?

Solve the equation on the interval [0, 2π). - cos⁡x+2cos⁡xsin⁡x=0\cos x + 2 \cos x \sin x = 0cosx+2cosxsinx=0

Express the sum or difference as a product. - cos⁡7x2+cos⁡5x2\cos \frac { 7 x } { 2 } + \cos \frac { 5 x } { 2 }cos27x​+cos25x​

Express the product as a sum or difference. - sin⁡6xsin⁡2x\sin 6 x \sin 2 xsin6xsin2x

Use Sum and Difference Formulas for Tangents Find the exact value by using a difference identity. - tan⁡255∘\tan 255 ^ { \circ }tan255∘

Use the given information to find the exact value of the trigonometric function. - csc⁡θ=−65,tan⁡θ>0\csc \theta = - \frac { 6 } { 5 } , \tan \theta > 0 \quadcscθ=−56​,tanθ>0 Find cos⁡θ2\cos \frac { \theta } { 2 }cos2θ​ .

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

Use Identities to Solve Trigonometric Equations Solve the equation on the interval [0, 2π). - sin⁡2x+sin⁡x=0\sin 2 x + \sin x = 0sin2x+sinx=0

Use Identities to Solve Trigonometric Equations Solve the equation on the interval [0, 2π). - cos⁡(x+π3)+cos⁡(x−π3)=1\cos \left( x + \frac { \pi } { 3 } \right) + \cos \left( x - \frac { \pi } { 3 } \right) = 1cos(x+3π​)+cos(x−3π​)=1

Use the given information to find the exact value of the trigonometric function. - sin⁡θ=−35,θ\sin \theta = - \frac { 3 } { 5 } , \quad \thetasinθ=−53​,θ lies in quadrant IV Find sin⁡θ2\sin \frac { \theta } { 2 }sin2θ​ .

Write the expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. - 2sin⁡75∘cos⁡75∘2 \sin 75 ^ { \circ } \cos 75 ^ { \circ }2sin75∘cos75∘

Express the sum or difference as a product. - sin⁡9x2+sin⁡5x2\sin \frac { 9 x } { 2 } + \sin \frac { 5 x } { 2 }sin29x​+sin25x​

Use Identities to Solve Trigonometric Equations Solve the equation on the interval [0, 2π). - sec⁡x2=cos⁡x2\sec \frac { x } { 2 } = \cos \frac { x } { 2 }sec2x​=cos2x​

Use Sum and Difference Formulas for Tangents Find the exact value by using a difference identity. - tan⁡285∘\tan 285 ^ { \circ }tan285∘

Complete the identity. - sin⁡2x+tan⁡2x+cos⁡2x=?\sin ^ { 2 } x + \tan ^ { 2 } x + \cos ^ { 2 } x = ?sin2x+tan2x+cos2x=?

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

Use the Power-Reducing Formulas - 3sin⁡2xcos⁡2x3 \sin ^ { 2 } x \cos ^ { 2 } x3sin2xcos2x