Question 1

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

Accepted Answer

A)  $0 , \pi$ 
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 
A)  $0 , \pi$ 
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

Question 2

Use Sum and Difference Formulas for Cosines and Sines
-$\sin 195 ^ { \circ } \cos 75 ^ { \circ } - \cos 195 ^ { \circ } \sin 75 ^ { \circ }$

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $- \frac { 1 } { 2 }$ 
C)  $\frac { 13 } { 4 }$ 
D)  $- \frac { \sqrt { 3 } } { 2 }$ 
A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $- \frac { 1 } { 2 }$ 
C)  $\frac { 13 } { 4 }$ 
D)  $- \frac { \sqrt { 3 } } { 2 }$

Question 3

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

Accepted Answer

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

Question 4

Use substitution to determine whether the given x-value is a solution of the equation.
Find All Solutions of a Trigonometric Equation
-$\cos 2 x = - \sqrt { 2 } , \quad x = \frac { 3 \pi } { 4 }$

Accepted Answer

A)  Yes 
B)  $\mathrm { No }$ 
A)  Yes 
B)  $\mathrm { No }$

Question 5

Complete the identity.
-$\csc ^ { 2 } x \sec x =$ ?

Accepted Answer

A)  $\sec x + \csc x \cot x$ 
B)  $\sec x - \csc x \cot x$ 
C)  $\csc x \cot x - \sec x$ 
D)  $\sec x + \csc x$ 
A)  $\sec x + \csc x \cot x$ 
B)  $\sec x - \csc x \cot x$ 
C)  $\csc x \cot x - \sec x$ 
D)  $\sec x + \csc x$

Question 6

Complete the identity.
-$\sin ^ { 4 } x - \cos ^ { 4 } x =$ ?

Accepted Answer

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

Question 7

Use the graph to complete the identity.
-$\csc x + \tan ^ { 2 } x \csc x ; \cos x$ and $\sin x$

Accepted Answer

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

Question 8

Use the figure to find the exact value of the trigonometric function.
-$\sin \theta = \frac { 12 } { 13 } , \theta$ lies in quadrant $\mathrm { I } \quad$ Find $\cos 2 \theta$.

Accepted Answer

A)  $- \frac { 119 } { 169 }$ 
B)  $\frac { 119 } { 169 }$ 
C)  $\frac { 120 } { 169 }$ 
D)  $- \frac { 121 } { 169 }$ 
A)  $- \frac { 119 } { 169 }$ 
B)  $\frac { 119 } { 169 }$ 
C)  $\frac { 120 } { 169 }$ 
D)  $- \frac { 121 } { 169 }$

Question 9

Solve Apps: Trigonometric Equations
-The weekly sales in thousands of items of a product has a seasonal sales record approximated by $\mathrm { n } = 84.37 + 15 \sin \frac { \pi \mathrm { t } } { 24 }$ ( $\mathrm { t }$ is time in weeks with $\mathrm { t } = 1$ referring to the first week in the year). During which week(s) will the sales equal 91,870 items?

Accepted Answer

A)  week 4 , week 20 , and week 52 
B)  week 30 and week 47 
C)  week 4 and week 47 
D)  week 21 and week 30 
A)  week 4 , week 20 , and week 52 
B)  week 30 and week 47 
C)  week 4 and week 47 
D)  week 21 and week 30

Question 10

Use the given information to find the exact value of the expression.
-$\sin \alpha = \frac { 20 } { 29 } , \alpha$ lies in quadrant $\mathrm { I }$, and $\cos \beta = \frac { 4 } { 5 } , \beta$ lies in quadrant I $\quad$ Find $\cos ( \alpha + \beta )$

Accepted Answer

A)  $\frac { 24 } { 145 }$ 
B)  $\frac { 144 } { 145 }$ 
C)  $\frac { 17 } { 145 }$ 
D)  $\frac { 143 } { 145 }$ 
A)  $\frac { 24 } { 145 }$ 
B)  $\frac { 144 } { 145 }$ 
C)  $\frac { 17 } { 145 }$ 
D)  $\frac { 143 } { 145 }$

Question 11

Use Sum and Difference Formulas for Tangents
Find the exact value by using a difference identity.
-$$\frac { \tan 175 ^ { \circ } - \tan 55 ^ { \circ } } { 1 + \tan 175 ^ { \circ } \tan 55 ^ { \circ } }$$

Accepted Answer

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

Question 12

Use The Half-Angle Formulas
-$\tan 165 ^ { \circ }$

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 13

Complete the identity.
-$\frac { ( \csc x + 1 ) ( \csc x - 1 ) } { \cot ^ { 2 } x } = ?$

Accepted Answer

A)  1 
B)  2 
C)  $- 1$ 
D)  0 )0 
A)  1 
B)  2 
C)  $- 1$ 
D)  0 )0

Question 14

Complete the identity.
-$\sin x ( \sin 2 x + \sin 4 x ) =$ ?

Accepted Answer

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

Question 15

Use the graph to complete the identity.
-$\frac { ( \sec x + \tan x ) ( \sec x - \tan x ) } { \sec x } =$ ?

Accepted Answer

A)  $\cos x$ 
B)  $\sin x$ 
C)  $\csc x$ 
D)  $\sec x$ 
A)  $\cos x$ 
B)  $\sin x$ 
C)  $\csc x$ 
D)  $\sec x$

Question 16

Solve the equation on the interval [0, 2π).
-$\sin 4 x = \frac { \sqrt { 3 } } { 2 }$

Accepted Answer

A)  $\frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } , \frac { 19 \pi } { 12 }$ 
B)  $\frac { \pi } { 4 } , \frac { 5 \pi } { 4 }$ 
C)  $0 , \frac { \pi } { 4 } , \pi$ 
D)  0 
A)  $\frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } , \frac { 19 \pi } { 12 }$ 
B)  $\frac { \pi } { 4 } , \frac { 5 \pi } { 4 }$ 
C)  $0 , \frac { \pi } { 4 } , \pi$ 
D)  0

Question 17

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

Accepted Answer

A)  $\frac { 1 } { 2 } ( \cos 3 x + \cos 4 x )$ 
B)  $\frac { 1 } { 4 } ( \cos 8 x - \sin 6 x )$ 
C)  $\frac { 1 } { 4 } \cos ^ { 2 } 7 x$ 
D)  $\frac { 1 } { 2 } ( \cos 4 x - \sin 3 x )$ 
A)  $\frac { 1 } { 2 } ( \cos 3 x + \cos 4 x )$ 
B)  $\frac { 1 } { 4 } ( \cos 8 x - \sin 6 x )$ 
C)  $\frac { 1 } { 4 } \cos ^ { 2 } 7 x$ 
D)  $\frac { 1 } { 2 } ( \cos 4 x - \sin 3 x )$

Question 18

Complete the identity.
-An airplane flying faster than the speed of sound creates sound waves that form a cone. If $\alpha$ is the vertex angle of the cone and $m$ is the Mach number for the speed of the plane, then $\sin \frac { \alpha } { 2 } = \frac { 1 } { m } ( m > 1 )$. Find the Mach number to the nearest tenth if $\alpha = 90 ^ { \circ }$.

Accepted Answer

A)  $1.4$ 
B)  $1.0$ 
C)  $1.7$ 
D)  $2.4$ 
A)  $1.4$ 
B)  $1.0$ 
C)  $1.7$ 
D)  $2.4$

Question 19

Complete the identity.
-$\sec ^ { 4 } x + \sec ^ { 2 } x \tan ^ { 2 } x - 2 \tan ^ { 4 } x =$ ?

Accepted Answer

A)  $3 \sec ^ { 4 } x - 2$ 
B)  $4 \sec ^ { 4 } x$ 
C)  $\sec ^ { 4 } x + 2$ 
D)  $\tan ^ { 2 } x - 1$ 
A)  $3 \sec ^ { 4 } x - 2$ 
B)  $4 \sec ^ { 4 } x$ 
C)  $\sec ^ { 4 } x + 2$ 
D)  $\tan ^ { 2 } x - 1$

Question 20

Use the graph to complete the identity.
-$( \sec x + \csc x ) ( \sin x + \cos x ) - 2 - \cot x ; \tan x$

Accepted Answer

A)  $\tan x$ 
B)  $2 + \tan x$ 
C)  $2 \tan x$ 
D)  0 
A)  $\tan x$ 
B)  $2 + \tan x$ 
C)  $2 \tan x$ 
D)  0

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

Use Sum and Difference Formulas for Cosines and Sines - $\sin 195 ^ { \circ } \cos 75 ^ { \circ } - \cos 195 ^ { \circ } \sin 75 ^ { \circ }$

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

Use substitution to determine whether the given x-value is a solution of the equation. Find All Solutions of a Trigonometric Equation - $\cos 2 x = - \sqrt { 2 } , \quad x = \frac { 3 \pi } { 4 }$

Complete the identity. - $\csc ^ { 2 } x \sec x =$ ?

Complete the identity. - $\sin ^ { 4 } x - \cos ^ { 4 } x =$ ?

Use the graph to complete the identity. - $\csc x + \tan ^ { 2 } x \csc x ; \cos x$ and $\sin x$

Use the figure to find the exact value of the trigonometric function. - $\sin \theta = \frac { 12 } { 13 } , \theta$ lies in quadrant $\mathrm { I } \quad$ Find $\cos 2 \theta$ .

Use the given information to find the exact value of the expression. - $\sin \alpha = \frac { 20 } { 29 } , \alpha$ lies in quadrant $\mathrm { I }$ , and $\cos \beta = \frac { 4 } { 5 } , \beta$ lies in quadrant I $\quad$ Find $\cos ( \alpha + \beta )$

Use Sum and Difference Formulas for Tangents Find the exact value by using a difference identity. - $\frac { \tan 175 ^ { \circ } - \tan 55 ^ { \circ } } { 1 + \tan 175 ^ { \circ } \tan 55 ^ { \circ } }$

Use The Half-Angle Formulas - $\tan 165 ^ { \circ }$

Complete the identity. - $\frac { ( \csc x + 1 ) ( \csc x - 1 ) } { \cot ^ { 2 } x } = ?$

Complete the identity. - $\sin x ( \sin 2 x + \sin 4 x ) =$ ?

Use the graph to complete the identity. - $\frac { ( \sec x + \tan x ) ( \sec x - \tan x ) } { \sec x } =$ ? $Use the graph to complete the identity. - \frac { ( \sec x + \tan x ) ( \sec x - \tan x ) } { \sec x } = ?$

Solve the equation on the interval [0, 2π). - $\sin 4 x = \frac { \sqrt { 3 } } { 2 }$

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

Complete the identity. - $\sec ^ { 4 } x + \sec ^ { 2 } x \tan ^ { 2 } x - 2 \tan ^ { 4 } x =$ ?

Use the graph to complete the identity. - $( \sec x + \csc x ) ( \sin x + \cos x ) - 2 - \cot x ; \tan x$

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 Identities to Solve Trigonometric Equations Solve the equation on the interval [0, 2π). - tan⁡2x−tan⁡x=0\tan 2 x - \tan x = 0tan2x−tanx=0

Use Sum and Difference Formulas for Cosines and Sines - sin⁡195∘cos⁡75∘−cos⁡195∘sin⁡75∘\sin 195 ^ { \circ } \cos 75 ^ { \circ } - \cos 195 ^ { \circ } \sin 75 ^ { \circ }sin195∘cos75∘−cos195∘sin75∘

Use the Power-Reducing Formulas - cos⁡3x\cos ^ { 3 } xcos3x

Use substitution to determine whether the given x-value is a solution of the equation. Find All Solutions of a Trigonometric Equation - cos⁡2x=−2,x=3π4\cos 2 x = - \sqrt { 2 } , \quad x = \frac { 3 \pi } { 4 }cos2x=−2​,x=43π​

Complete the identity. - csc⁡2xsec⁡x=\csc ^ { 2 } x \sec x =csc2xsecx= ?

Complete the identity. - sin⁡4x−cos⁡4x=\sin ^ { 4 } x - \cos ^ { 4 } x =sin4x−cos4x= ?

Use the graph to complete the identity. - csc⁡x+tan⁡2xcsc⁡x;cos⁡x\csc x + \tan ^ { 2 } x \csc x ; \cos xcscx+tan2xcscx;cosx and sin⁡x\sin xsinx

Use the figure to find the exact value of the trigonometric function. - sin⁡θ=1213,θ\sin \theta = \frac { 12 } { 13 } , \thetasinθ=1312​,θ lies in quadrant I\mathrm { I } \quadI Find cos⁡2θ\cos 2 \thetacos2θ .

Use Sum and Difference Formulas for Tangents Find the exact value by using a difference identity. - tan⁡175∘−tan⁡55∘1+tan⁡175∘tan⁡55∘\frac { \tan 175 ^ { \circ } - \tan 55 ^ { \circ } } { 1 + \tan 175 ^ { \circ } \tan 55 ^ { \circ } }1+tan175∘tan55∘tan175∘−tan55∘​

Use The Half-Angle Formulas - tan⁡165∘\tan 165 ^ { \circ }tan165∘

Complete the identity. - (csc⁡x+1)(csc⁡x−1)cot⁡2x=?\frac { ( \csc x + 1 ) ( \csc x - 1 ) } { \cot ^ { 2 } x } = ?cot2x(cscx+1)(cscx−1)​=?

Complete the identity. - sin⁡x(sin⁡2x+sin⁡4x)=\sin x ( \sin 2 x + \sin 4 x ) =sinx(sin2x+sin4x)= ?

Use the graph to complete the identity. - (sec⁡x+tan⁡x)(sec⁡x−tan⁡x)sec⁡x=\frac { ( \sec x + \tan x ) ( \sec x - \tan x ) } { \sec x } =secx(secx+tanx)(secx−tanx)​= ?

Solve the equation on the interval [0, 2π). - sin⁡4x=32\sin 4 x = \frac { \sqrt { 3 } } { 2 }sin4x=23​​

Express the product as a sum or difference. - cos⁡7x2cos⁡x2\cos \frac { 7 x } { 2 } \cos \frac { x } { 2 }cos27x​cos2x​

Complete the identity. - sec⁡4x+sec⁡2xtan⁡2x−2tan⁡4x=\sec ^ { 4 } x + \sec ^ { 2 } x \tan ^ { 2 } x - 2 \tan ^ { 4 } x =sec4x+sec2xtan2x−2tan4x= ?

Use the graph to complete the identity. - (sec⁡x+csc⁡x)(sin⁡x+cos⁡x)−2−cot⁡x;tan⁡x( \sec x + \csc x ) ( \sin x + \cos x ) - 2 - \cot x ; \tan x(secx+cscx)(sinx+cosx)−2−cotx;tanx