Question 1

Use a calculator to solve the equation on the interval [0, 2π). Round the answer to two decimal places.
-$\cos 3 x = - \cos x$

Accepted Answer

A)  $0.79,1.57,2.36,3.93,4.71,5.50$ 
B)  $0,0.79,2.36,3.14,3.93,5.50$ 
C)  $0,1.57,3.14,4.71$ 
D)  $1.57,4.71$ 
A)  $0.79,1.57,2.36,3.93,4.71,5.50$ 
B)  $0,0.79,2.36,3.14,3.93,5.50$ 
C)  $0,1.57,3.14,4.71$ 
D)  $1.57,4.71$

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

Complete the identity.
-$\cos \left( x - \frac { 11 \pi } { 6 } \right) =$ ?

Accepted Answer

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

Question 5

Find the exact value under the given conditions.
-$\tan \alpha = \frac { 5 } { 12 } , \pi < \alpha < \frac { 3 \pi } { 2 } ; \quad \cos \beta = - \frac { 3 } { 5 } , \frac { \pi } { 2 } < \beta < \pi \quad$ Find $\tan ( \alpha + \beta )$

Accepted Answer

A)  $- \frac { 33 } { 56 }$ 
B)  $\frac { 56 } { 65 }$ 
C)  $\frac { 2 } { 7 }$ 
D)  $- \frac { 33 } { 65 }$ 
A)  $- \frac { 33 } { 56 }$ 
B)  $\frac { 56 } { 65 }$ 
C)  $\frac { 2 } { 7 }$ 
D)  $- \frac { 33 } { 65 }$

Question 6

Use the given information to find the exact value of the expression.
-$\cos \alpha = - \frac { 4 } { 5 } , \alpha$ lies in quadrant III, and $\sin \beta = \frac { \sqrt { 21 } } { 5 } , \beta$ lies in quadrant II $\quad$ Find $\cos ( \alpha + \beta )$

Accepted Answer

A)  $\frac { 8 + 3 \sqrt { 21 } } { 25 }$ 
B)  $\frac { - 8 - 3 \sqrt { 21 } } { 25 }$ 
C)  $\frac { 6 - 4 \sqrt { 21 } } { 25 }$ 
D)  $\frac { - 6 + 4 \sqrt { 21 } } { 25 }$ 
A)  $\frac { 8 + 3 \sqrt { 21 } } { 25 }$ 
B)  $\frac { - 8 - 3 \sqrt { 21 } } { 25 }$ 
C)  $\frac { 6 - 4 \sqrt { 21 } } { 25 }$ 
D)  $\frac { - 6 + 4 \sqrt { 21 } } { 25 }$

Question 7

Express the sum or difference as a product.
-$\sin 9 x + \sin 3 x$

Accepted Answer

A)  $2 \sin 6 x \cos 3 x$ 
B)  $2 \cos 6 x \sin 3 x$ 
C)  $2 \sin 6 x \sin 3 x$ 
D)  $2 \sin 12 x$ 
A)  $2 \sin 6 x \cos 3 x$ 
B)  $2 \cos 6 x \sin 3 x$ 
C)  $2 \sin 6 x \sin 3 x$ 
D)  $2 \sin 12 x$

Question 8

Use the given information to find the exact value of the trigonometric function.
-$\sec \theta = 4 , \theta$ lies in quadrant I $\quad$ Find $\cos \frac { \theta } { 2 }$

Accepted Answer

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

Question 9

Use the given information to find the exact value of the expression.
-$\sin \alpha = \frac { 24 } { 25 } , \alpha$ lies in quadrant II, and $\cos \beta = \frac { 2 } { 5 } , \beta$ lies in quadrant $\mathrm { I } \quad$ Find $\cos ( \alpha - \beta )$.

Accepted Answer

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

Question 10

Use the Power-Reducing Formulas
-$8 \cos ^ { 2 } x$

Accepted Answer

A)  $4 + 4 \cos 2 x$ 
B)  $4 - 4 \cos 2 x$ 
C)  $1 + \cos 2 x$ 
D)  $16 \cos x$ 
A)  $4 + 4 \cos 2 x$ 
B)  $4 - 4 \cos 2 x$ 
C)  $1 + \cos 2 x$ 
D)  $16 \cos x$

Question 11

Verify the identity.
-$$\csc u - \sin u = \cos u \cot u$$

Accepted Answer

The answer of Verify the identity.
-\[\csc u - \sin u...

Question 12

Use the given information to find the exact value of the trigonometric function.
-$\sin \theta = \frac { 1 } { 4 } , \tan \theta > 0 \quad$ Find $\cos \frac { \theta } { 2 }$.

Accepted Answer

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

Question 13

Use the given information to find the exact value of the expression.
-$\sin \alpha = - \frac { 24 } { 25 } , \alpha$ lies in quadrant IV, and $\cos \beta = - \frac { \sqrt { 21 } } { 5 } , \beta$ lies in quadrant III Find $\sin ( \alpha - \beta )$.

Accepted Answer

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

Question 14

Use Identities to Solve Trigonometric Equations
Solve the equation on the interval [0, 2π).
-$\sin \left( x + \frac { 7 \pi } { 6 } \right) - \sin \left( x - \frac { 7 \pi } { 6 } \right) = 1$

Accepted Answer

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

Question 15

Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal.
-$\cos x - \cos x \sin x = \cos ^ { 3 } x$

Accepted Answer

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

Question 16

Find the exact value under the given conditions.
-$\cos \left( \frac { 5 \pi } { 3 } + \alpha \right) \cos \left( \frac { 5 \pi } { 3 } - \alpha \right) - \sin \left( \frac { 5 \pi } { 3 } + \alpha \right) \sin \left( \frac { 5 \pi } { 3 } - \alpha \right)$

Accepted Answer

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

Question 17

Use the figure to find the exact value of the trigonometric function.
-$\sin \theta = \frac { 4 } { 5 } , \theta$ lies in quadrant II $\quad$ Find $\tan 2 \theta$.

Accepted Answer

A)  $\frac { 24 } { 7 }$ 
B)  $- \frac { 24 } { 7 }$ 
C)  $\frac { 24 } { 25 }$ 
D)  $- \frac { 5 } { 7 }$ 
A)  $\frac { 24 } { 7 }$ 
B)  $- \frac { 24 } { 7 }$ 
C)  $\frac { 24 } { 25 }$ 
D)  $- \frac { 5 } { 7 }$

Question 18

Verify the identity.
-$$\cos ( \alpha - \beta ) - \cos ( \alpha + \beta ) = 2 \sin \alpha \sin \beta$$

Accepted Answer

The answer of Verify the identity.
-\[\cos ( \alpha - \beta...

Question 19

Verify the identity.
-$$\frac { \cos x + \cos y } { \sin x - \sin y } = \cot \frac { x - y } { 2 }$$

Accepted Answer

The answer of Verify the identity.
-\[\frac { \cos x +...

Question 20

Use the graph to complete the identity.
-$$\frac { 1 + \cos x } { \sin x } + \frac { \sin x } { 1 + \cos x } = ?$$

Accepted Answer

A)  $2 \csc x$ 
B)  $2 \sec x$ 
C)  $2 \sin x$ 
D)  $2 \cos x$ Rewrite the expression in terms of the given function or functions. 
A)  $2 \csc x$ 
B)  $2 \sec x$ 
C)  $2 \sin x$ 
D)  $2 \cos x$ Rewrite the expression in terms of the given function or functions.

Use a calculator to solve the equation on the interval [0, 2π). Round the answer to two decimal places. - $\cos 3 x = - \cos x$

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

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

Complete the identity. - $\cos \left( x - \frac { 11 \pi } { 6 } \right) =$ ?

Find the exact value under the given conditions. - $\tan \alpha = \frac { 5 } { 12 } , \pi < \alpha < \frac { 3 \pi } { 2 } ; \quad \cos \beta = - \frac { 3 } { 5 } , \frac { \pi } { 2 } < \beta < \pi \quad$ Find $\tan ( \alpha + \beta )$

Use the given information to find the exact value of the expression. - $\cos \alpha = - \frac { 4 } { 5 } , \alpha$ lies in quadrant III, and $\sin \beta = \frac { \sqrt { 21 } } { 5 } , \beta$ lies in quadrant II $\quad$ Find $\cos ( \alpha + \beta )$

Express the sum or difference as a product. - $\sin 9 x + \sin 3 x$

Use the given information to find the exact value of the trigonometric function. - $\sec \theta = 4 , \theta$ lies in quadrant I $\quad$ Find $\cos \frac { \theta } { 2 }$

Use the given information to find the exact value of the expression. - $\sin \alpha = \frac { 24 } { 25 } , \alpha$ lies in quadrant II, and $\cos \beta = \frac { 2 } { 5 } , \beta$ lies in quadrant $\mathrm { I } \quad$ Find $\cos ( \alpha - \beta )$ .

Use the Power-Reducing Formulas - $8 \cos ^ { 2 } x$

Verify the identity. - $\csc u - \sin u = \cos u \cot u$

Use the given information to find the exact value of the trigonometric function. - $\sin \theta = \frac { 1 } { 4 } , \tan \theta > 0 \quad$ Find $\cos \frac { \theta } { 2 }$ .

Use the given information to find the exact value of the expression. - $\sin \alpha = - \frac { 24 } { 25 } , \alpha$ lies in quadrant IV, and $\cos \beta = - \frac { \sqrt { 21 } } { 5 } , \beta$ lies in quadrant III Find $\sin ( \alpha - \beta )$ .

Use Identities to Solve Trigonometric Equations Solve the equation on the interval [0, 2π). - $\sin \left( x + \frac { 7 \pi } { 6 } \right) - \sin \left( x - \frac { 7 \pi } { 6 } \right) = 1$

Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal. - $\cos x - \cos x \sin x = \cos ^ { 3 } x$

Find the exact value under the given conditions. - $\cos \left( \frac { 5 \pi } { 3 } + \alpha \right) \cos \left( \frac { 5 \pi } { 3 } - \alpha \right) - \sin \left( \frac { 5 \pi } { 3 } + \alpha \right) \sin \left( \frac { 5 \pi } { 3 } - \alpha \right)$

Use the figure to find the exact value of the trigonometric function. - $\sin \theta = \frac { 4 } { 5 } , \theta$ lies in quadrant II $\quad$ Find $\tan 2 \theta$ .

Verify the identity. - $\cos ( \alpha - \beta ) - \cos ( \alpha + \beta ) = 2 \sin \alpha \sin \beta$

Verify the identity. - $\frac { \cos x + \cos y } { \sin x - \sin y } = \cot \frac { x - y } { 2 }$

Use the graph to complete the identity. - $\frac { 1 + \cos x } { \sin x } + \frac { \sin x } { 1 + \cos x } = ?$ $Use the graph to complete the identity. - \frac { 1 + \cos x } { \sin x } + \frac { \sin x } { 1 + \cos x } = ?$

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 a calculator to solve the equation on the interval [0, 2π). Round the answer to two decimal places. - cos⁡3x=−cos⁡x\cos 3 x = - \cos xcos3x=−cosx

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

Solve the equation on the interval [0, 2π). - sin⁡2x−cos⁡2x=0\sin ^ { 2 } x - \cos ^ { 2 } x = 0sin2x−cos2x=0

Complete the identity. - cos⁡(x−11π6)=\cos \left( x - \frac { 11 \pi } { 6 } \right) =cos(x−611π​)= ?

Express the sum or difference as a product. - sin⁡9x+sin⁡3x\sin 9 x + \sin 3 xsin9x+sin3x

Use the given information to find the exact value of the trigonometric function. - sec⁡θ=4,θ\sec \theta = 4 , \thetasecθ=4,θ lies in quadrant I \quad Find cos⁡θ2\cos \frac { \theta } { 2 }cos2θ​

Use the Power-Reducing Formulas - 8cos⁡2x8 \cos ^ { 2 } x8cos2x

Verify the identity. - csc⁡u−sin⁡u=cos⁡ucot⁡u\csc u - \sin u = \cos u \cot ucscu−sinu=cosucotu

Use the given information to find the exact value of the trigonometric function. - sin⁡θ=14,tan⁡θ>0\sin \theta = \frac { 1 } { 4 } , \tan \theta > 0 \quadsinθ=41​,tanθ>0 Find cos⁡θ2\cos \frac { \theta } { 2 }cos2θ​ .

Use Identities to Solve Trigonometric Equations Solve the equation on the interval [0, 2π). - sin⁡(x+7π6)−sin⁡(x−7π6)=1\sin \left( x + \frac { 7 \pi } { 6 } \right) - \sin \left( x - \frac { 7 \pi } { 6 } \right) = 1sin(x+67π​)−sin(x−67π​)=1

Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal. - cos⁡x−cos⁡xsin⁡x=cos⁡3x\cos x - \cos x \sin x = \cos ^ { 3 } xcosx−cosxsinx=cos3x

Use the figure to find the exact value of the trigonometric function. - sin⁡θ=45,θ\sin \theta = \frac { 4 } { 5 } , \thetasinθ=54​,θ lies in quadrant II \quad Find tan⁡2θ\tan 2 \thetatan2θ .

Verify the identity. - cos⁡(α−β)−cos⁡(α+β)=2sin⁡αsin⁡β\cos ( \alpha - \beta ) - \cos ( \alpha + \beta ) = 2 \sin \alpha \sin \betacos(α−β)−cos(α+β)=2sinαsinβ

Verify the identity. - cos⁡x+cos⁡ysin⁡x−sin⁡y=cot⁡x−y2\frac { \cos x + \cos y } { \sin x - \sin y } = \cot \frac { x - y } { 2 }sinx−sinycosx+cosy​=cot2x−y​

Use the graph to complete the identity. - 1+cos⁡xsin⁡x+sin⁡x1+cos⁡x=?\frac { 1 + \cos x } { \sin x } + \frac { \sin x } { 1 + \cos x } = ?sinx1+cosx​+1+cosxsinx​=?