Question 1

Complete the identity.
-$\sin 4 x \sin 6 x \cos 4 x \cos 6 x =$ ?

Accepted Answer

A)  $\frac { \cos ^ { 2 } 2 x - \cos ^ { 2 } 10 x } { 4 }$ 
B)  $\frac { \sin ^ { 2 } 48 x } { 4 }$ 
C)  $\frac { \cos ^ { 2 } 10 x + \cos ^ { 2 } 2 x } { 4 }$ 
D)  $\cos ^ { 2 } 48 x$ 
A)  $\frac { \cos ^ { 2 } 2 x - \cos ^ { 2 } 10 x } { 4 }$ 
B)  $\frac { \sin ^ { 2 } 48 x } { 4 }$ 
C)  $\frac { \cos ^ { 2 } 10 x + \cos ^ { 2 } 2 x } { 4 }$ 
D)  $\cos ^ { 2 } 48 x$

Question 2

Verify the identity.
-$$\cos 4 \theta = 2 \cos ^ { 2 } ( 2 \theta ) - 1$$

Accepted Answer

The answer of Verify the identity.
-\[\cos 4 \theta = 2...

Question 3

Use substitution to determine whether the given x-value is a solution of the equation.
Find All Solutions of a Trigonometric Equation
-$\cos x + 1 = \sin x , \quad x = \frac { - 3 \pi } { 4 }$

Accepted Answer

A)  Yes 
B)  $\mathrm { No }$ Find all solutions of the equation. 
A)  Yes 
B)  $\mathrm { No }$ Find all solutions of the equation.

Question 4

Complete the identity.
-$8 \sin x \cos ^ { 3 } x + 8 \sin ^ { 3 } x \cos x =$ ?

Accepted Answer

A)  $4 \sin 2 x$ 
B)  $4 \cos 2 x$ 
C)  $4 \sin x$ 
D)  $4 \cos x$ 
A)  $4 \sin 2 x$ 
B)  $4 \cos 2 x$ 
C)  $4 \sin x$ 
D)  $4 \cos x$

Question 5

Complete the identity.
-$\frac { \cos ^ { 2 } x - \sin ^ { 2 } x } { 1 - \tan ^ { 2 } x } = ?$

Accepted Answer

A)  $\cos ^ { 2 } x$ 
B)  $\sin ^ { 2 } x$ 
C)  1 
D)  $- 1$ 
A)  $\cos ^ { 2 } x$ 
B)  $\sin ^ { 2 } x$ 
C)  1 
D)  $- 1$

Question 6

Express the sum or difference as a product.
-$\sin 75 ^ { \circ } + \sin 15 ^ { \circ }$

Accepted Answer

A)  $\frac { \sqrt { 6 } } { 2 }$ 
B)  $\frac { \sqrt { 6 } } { 4 }$ 
C)  $\frac { \sqrt { 3 } } { 2 }$ 
D)  $- \frac { \sqrt { 6 } } { 2 }$ 
A)  $\frac { \sqrt { 6 } } { 2 }$ 
B)  $\frac { \sqrt { 6 } } { 4 }$ 
C)  $\frac { \sqrt { 3 } } { 2 }$ 
D)  $- \frac { \sqrt { 6 } } { 2 }$

Question 7

Describe the graph using another equation.
-$y=\cos \left(x+\frac{\pi}{2}\right)-\cos \left(x-\frac{\pi}{2}\right)$

Accepted Answer

A)  $- 2 \sin x$ 
B)  $- 2 \cos x$ 
C)  $- \sin x$ 
D)  $- \cos x$ 
A)  $- 2 \sin x$ 
B)  $- 2 \cos x$ 
C)  $- \sin x$ 
D)  $- \cos x$

Question 8

Fill in the blank using the word product, sum, quotient, or difference.
-The formula $\sin \alpha - \sin \beta = 2 \sin \frac { \alpha - \beta } { 2 } \cos \frac { \alpha + \beta } { 2 }$ can be used to change aـــــــــــ of two sines into the ــــــــــــof a sine and a cosine expression.

Accepted Answer

A)  difference, product 
B)  sum, product 
C)  difference, quotient 
D)  sum, quotient 
A)  difference, product 
B)  sum, product 
C)  difference, quotient 
D)  sum, quotient

Question 9

Use the figure to find the exact value of the trigonometric function.
-Find $\cos 2 \theta$.

Accepted Answer

A)  $- \frac { 119 } { 169 }$ 
B)  $\frac { 119 } { 169 }$ 
C)  $\frac { 120 } { 169 }$ 
D)  $- \frac { 9 } { 13 }$ 
A)  $- \frac { 119 } { 169 }$ 
B)  $\frac { 119 } { 169 }$ 
C)  $\frac { 120 } { 169 }$ 
D)  $- \frac { 9 } { 13 }$

Question 10

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

Accepted Answer

A)  $\frac { 3 } { 4 } \sin x - \frac { 1 } { 4 } \sin 3 x$ 
B)  $\frac { 3 } { 4 } \sin x + \frac { 1 } { 2 } \sin 2 x - \frac { 1 } { 4 } \sin 3 x$ 
C)  $\frac { 3 } { 4 } \sin x + \frac { 1 } { 4 } \sin 2 x$ 
D)  $\frac { 3 } { 4 } \sin x$ 
A)  $\frac { 3 } { 4 } \sin x - \frac { 1 } { 4 } \sin 3 x$ 
B)  $\frac { 3 } { 4 } \sin x + \frac { 1 } { 2 } \sin 2 x - \frac { 1 } { 4 } \sin 3 x$ 
C)  $\frac { 3 } { 4 } \sin x + \frac { 1 } { 4 } \sin 2 x$ 
D)  $\frac { 3 } { 4 } \sin x$

Question 11

Complete the identity.
-$$\frac { \sin 3 x + \sin 7 x } { \cos 3 x + \cos 7 x } = ?$$

Accepted Answer

A)  $\tan 5 x$ 
B)  $\tan 3 x + \tan 7 x$ 
C)  $2 \tan 5 x \tan 2 x$ 
D)  $\tan 5 x \cot 2 x$ 
A)  $\tan 5 x$ 
B)  $\tan 3 x + \tan 7 x$ 
C)  $2 \tan 5 x \tan 2 x$ 
D)  $\tan 5 x \cot 2 x$

Question 12

Solve Apps: Trigonometric Equations
-The output voltage for an $\mathrm { AC }$ generator is approximated by $\mathrm { v } = 156 \cos \left( 120 \pi \mathrm { t } - \frac { \pi } { 3 } \right)$. Find the smallest positive value of $t$ for which the output is 125 volts.

Accepted Answer

A)  $0.0045$ second 
B)  $0.0035$ second 
C)  $0.0090$ second 
D)  $0.0100$ second 
A)  $0.0045$ second 
B)  $0.0035$ second 
C)  $0.0090$ second 
D)  $0.0100$ second

Question 13

Verify the identity.
-$$\frac { \sin \alpha - \sin \beta } { \sin \alpha + \sin \beta } = \tan \frac { \alpha - \beta } { 2 } \cot \frac { \alpha + \beta } { 2 }$$

Accepted Answer

The answer of Verify the identity.
-\[\frac { \sin \alpha -...

Question 14

Verify the identity.
-$$\sin 4 t = 2 \sin 2 t \cos 2 t$$

Accepted Answer

The answer of Verify the identity.
-\[\sin 4 t = 2...

Question 15

Complete the identity.
-$$\frac { \sin x } { \cos x } + \frac { \cos x } { \sin x } = ?$$

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

Write the expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression.
-$$\frac { 2 \tan \frac { 7 \pi } { 12 } } { 1 - \tan ^ { 2 } \frac { 7 \pi } { 12 } }$$

Accepted Answer

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

Question 18

Complete the identity.
-$$1 - \frac { \sin ^ { 2 } x } { 1 + \cos x } = \text { ? }$$

Accepted Answer

A)  $\cos x$ 
B)  0 
C)  $\tan x$ 
D)  $\cot x$ 
A)  $\cos x$ 
B)  0 
C)  $\tan x$ 
D)  $\cot x$

Question 19

Use Sum and Difference Formulas for Cosines and Sines
-$\cos \frac { 7 \pi } { 12 } \sin \frac { 5 \pi } { 12 } - \cos \frac { 5 \pi } { 12 } \sin \frac { 7 \pi } { 12 }$

Accepted Answer

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

Question 20

Solve the equation on the interval [0, 2π).
-$\cos 2 x = \frac { \sqrt { 2 } } { 2 }$

Accepted Answer

A)  $\frac { \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 15 \pi } { 8 }$ 
B)  $0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
C)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
D)  no solution 
A)  $\frac { \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 15 \pi } { 8 }$ 
B)  $0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
C)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
D)  no solution

Complete the identity. - $\sin 4 x \sin 6 x \cos 4 x \cos 6 x =$ ?

Verify the identity. - $\cos 4 \theta = 2 \cos ^ { 2 } ( 2 \theta ) - 1$

Use substitution to determine whether the given x-value is a solution of the equation. Find All Solutions of a Trigonometric Equation - $\cos x + 1 = \sin x , \quad x = \frac { - 3 \pi } { 4 }$

Complete the identity. - $8 \sin x \cos ^ { 3 } x + 8 \sin ^ { 3 } x \cos x =$ ?

Complete the identity. - $\frac { \cos ^ { 2 } x - \sin ^ { 2 } x } { 1 - \tan ^ { 2 } x } = ?$

Express the sum or difference as a product. - $\sin 75 ^ { \circ } + \sin 15 ^ { \circ }$

Describe the graph using another equation. - $y=\cos \left(x+\frac{\pi}{2}\right)-\cos \left(x-\frac{\pi}{2}\right)$ $Describe the graph using another equation. - y=\cos \left(x+\frac{\pi}{2}\right)-\cos \left(x-\frac{\pi}{2}\right)$

Use the figure to find the exact value of the trigonometric function. -Find $\cos 2 \theta$ . $Use the figure to find the exact value of the trigonometric function. -Find \cos 2 \theta .$

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

Complete the identity. - $\frac { \sin 3 x + \sin 7 x } { \cos 3 x + \cos 7 x } = ?$

Solve Apps: Trigonometric Equations -The output voltage for an $\mathrm { AC }$ generator is approximated by $\mathrm { v } = 156 \cos \left( 120 \pi \mathrm { t } - \frac { \pi } { 3 } \right)$ . Find the smallest positive value of $t$ for which the output is 125 volts.

Verify the identity. - $\frac { \sin \alpha - \sin \beta } { \sin \alpha + \sin \beta } = \tan \frac { \alpha - \beta } { 2 } \cot \frac { \alpha + \beta } { 2 }$

Verify the identity. - $\sin 4 t = 2 \sin 2 t \cos 2 t$

Complete the identity. - $\frac { \sin x } { \cos x } + \frac { \cos x } { \sin x } = ?$

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

Write the expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. - $\frac { 2 \tan \frac { 7 \pi } { 12 } } { 1 - \tan ^ { 2 } \frac { 7 \pi } { 12 } }$

Complete the identity. - $1 - \frac { \sin ^ { 2 } x } { 1 + \cos x } = \text { ? }$

Use Sum and Difference Formulas for Cosines and Sines - $\cos \frac { 7 \pi } { 12 } \sin \frac { 5 \pi } { 12 } - \cos \frac { 5 \pi } { 12 } \sin \frac { 7 \pi } { 12 }$

Solve the equation on the interval [0, 2π). - $\cos 2 x = \frac { \sqrt { 2 } } { 2 }$

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

Complete the identity. - sin⁡4xsin⁡6xcos⁡4xcos⁡6x=\sin 4 x \sin 6 x \cos 4 x \cos 6 x =sin4xsin6xcos4xcos6x= ?

Verify the identity. - cos⁡4θ=2cos⁡2(2θ)−1\cos 4 \theta = 2 \cos ^ { 2 } ( 2 \theta ) - 1cos4θ=2cos2(2θ)−1

Use substitution to determine whether the given x-value is a solution of the equation. Find All Solutions of a Trigonometric Equation - cos⁡x+1=sin⁡x,x=−3π4\cos x + 1 = \sin x , \quad x = \frac { - 3 \pi } { 4 }cosx+1=sinx,x=4−3π​

Complete the identity. - 8sin⁡xcos⁡3x+8sin⁡3xcos⁡x=8 \sin x \cos ^ { 3 } x + 8 \sin ^ { 3 } x \cos x =8sinxcos3x+8sin3xcosx= ?

Complete the identity. - cos⁡2x−sin⁡2x1−tan⁡2x=?\frac { \cos ^ { 2 } x - \sin ^ { 2 } x } { 1 - \tan ^ { 2 } x } = ?1−tan2xcos2x−sin2x​=?

Express the sum or difference as a product. - sin⁡75∘+sin⁡15∘\sin 75 ^ { \circ } + \sin 15 ^ { \circ }sin75∘+sin15∘

Describe the graph using another equation. - y=cos⁡(x+π2)−cos⁡(x−π2)y=\cos \left(x+\frac{\pi}{2}\right)-\cos \left(x-\frac{\pi}{2}\right)y=cos(x+2π​)−cos(x−2π​)

Use the figure to find the exact value of the trigonometric function. -Find cos⁡2θ\cos 2 \thetacos2θ .

Use the Power-Reducing Formulas - sin⁡3x\sin ^ { 3 } xsin3x

Complete the identity. - sin⁡3x+sin⁡7xcos⁡3x+cos⁡7x=?\frac { \sin 3 x + \sin 7 x } { \cos 3 x + \cos 7 x } = ?cos3x+cos7xsin3x+sin7x​=?

Verify the identity. - sin⁡α−sin⁡βsin⁡α+sin⁡β=tan⁡α−β2cot⁡α+β2\frac { \sin \alpha - \sin \beta } { \sin \alpha + \sin \beta } = \tan \frac { \alpha - \beta } { 2 } \cot \frac { \alpha + \beta } { 2 }sinα+sinβsinα−sinβ​=tan2α−β​cot2α+β​

Verify the identity. - sin⁡4t=2sin⁡2tcos⁡2t\sin 4 t = 2 \sin 2 t \cos 2 tsin4t=2sin2tcos2t

Complete the identity. - sin⁡xcos⁡x+cos⁡xsin⁡x=?\frac { \sin x } { \cos x } + \frac { \cos x } { \sin x } = ?cosxsinx​+sinxcosx​=?

Solve the equation on the interval [0, 2π). - sin⁡2x+sin⁡x=0\sin ^ { 2 } x + \sin x = 0sin2x+sinx=0

Write the expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. - 2tan⁡7π121−tan⁡27π12\frac { 2 \tan \frac { 7 \pi } { 12 } } { 1 - \tan ^ { 2 } \frac { 7 \pi } { 12 } }1−tan2127π​2tan127π​​

Complete the identity. - 1−sin⁡2x1+cos⁡x= ? 1 - \frac { \sin ^ { 2 } x } { 1 + \cos x } = \text { ? }1−1+cosxsin2x​= ?

Use Sum and Difference Formulas for Cosines and Sines - cos⁡7π12sin⁡5π12−cos⁡5π12sin⁡7π12\cos \frac { 7 \pi } { 12 } \sin \frac { 5 \pi } { 12 } - \cos \frac { 5 \pi } { 12 } \sin \frac { 7 \pi } { 12 }cos127π​sin125π​−cos125π​sin127π​

Solve the equation on the interval [0, 2π). - cos⁡2x=22\cos 2 x = \frac { \sqrt { 2 } } { 2 }cos2x=22​​