Question 1

Use the Power-Reducing Formulas
-$\sin ^ { 4 } x$

Accepted Answer

A)  $\frac { 3 } { 8 } - \frac { 1 } { 2 } \cos 2 x + \frac { 1 } { 8 } \cos 4 x$ 
B)  $\frac { 3 } { 2 } - \frac { 3 } { 2 } \cos 2 x$ 
C)  $\frac { 3 } { 2 } - 2 \cos 2 x + \frac { 1 } { 2 } \cos 4 x$ 
D)  $\frac { 3 } { 8 } + \frac { 5 } { 8 } \cos 2 x$ 
A)  $\frac { 3 } { 8 } - \frac { 1 } { 2 } \cos 2 x + \frac { 1 } { 8 } \cos 4 x$ 
B)  $\frac { 3 } { 2 } - \frac { 3 } { 2 } \cos 2 x$ 
C)  $\frac { 3 } { 2 } - 2 \cos 2 x + \frac { 1 } { 2 } \cos 4 x$ 
D)  $\frac { 3 } { 8 } + \frac { 5 } { 8 } \cos 2 x$

Question 2

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

Accepted Answer

A)  1 
B)  0 
C) $ - \sec ^ { 2 } x$ 
D)  $1 - \sin x$ 
A)  1 
B)  0 
C) $ - \sec ^ { 2 } x$ 
D)  $1 - \sin x$

Question 3

Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal.
-$$\frac { \cos 10 x - \cos 4 x } { \sin 10 x + \sin 4 x } = \text { ? }$$

Accepted Answer

A)  $- \tan 3 x$ 
B)  $\cot 4 x + \cot 10 x$ 
C)  $2 \tan 7 x \tan 3 x$ 
D)  $\tan 7 x$ 
A)  $- \tan 3 x$ 
B)  $\cot 4 x + \cot 10 x$ 
C)  $2 \tan 7 x \tan 3 x$ 
D)  $\tan 7 x$

Question 4

Use Sum and Difference Formulas for Cosines and Sines
-$\cos \left( 30 ^ { \circ } + 45 ^ { \circ } \right)$

Accepted Answer

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

Question 5

Solve the problem.
-The sound produced by touching each button on a touch-tone phone is described by $y = \sin 2 \pi l t + \sin 2 \pi h t$
where $l$ and $h$ are the low and high frequencies (cycles per second) in the figure shown.

Touch-Tone phone
Describe the sound emitted by touching the 0 key as a product of sines and cosines.

Accepted Answer

A)  $\mathrm { y } = 2 \sin ( 2277 \pi \mathrm { t } ) \cos ( 395 \pi \mathrm { t } )$ 
B)  $\mathrm { y } = 2 \sin ( 2418 \pi \mathrm { t } ) \cos ( 536 \pi \mathrm { t } )$ 
C)  $\mathrm { y } = 2 \sin ( 395 \pi \mathrm { t } ) \cos ( 2277 \pi \mathrm { t } )$ 
D)  $\mathrm { y } = 2 \sin ( 536 \pi \mathrm { t } ) \cos ( 2418 \pi \mathrm { t } )$ 
A)  $\mathrm { y } = 2 \sin ( 2277 \pi \mathrm { t } ) \cos ( 395 \pi \mathrm { t } )$ 
B)  $\mathrm { y } = 2 \sin ( 2418 \pi \mathrm { t } ) \cos ( 536 \pi \mathrm { t } )$ 
C)  $\mathrm { y } = 2 \sin ( 395 \pi \mathrm { t } ) \cos ( 2277 \pi \mathrm { t } )$ 
D)  $\mathrm { y } = 2 \sin ( 536 \pi \mathrm { t } ) \cos ( 2418 \pi \mathrm { t } )$

Question 6

Verify the identity.
-$$\cos \left( \frac { 3 \pi } { 2 } - \theta \right) = - \sin \theta$$

Accepted Answer

The answer of Verify the identity.
-\[\cos \left( \frac { 3...

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

A)  $1.05,3.14,5.24$ 
B)  $0,2.09,4.19$ 
C)  $0,2.09,3.14,4.19$ 
D)  $0,1.05,3.14,5.24$ 
A)  $1.05,3.14,5.24$ 
B)  $0,2.09,4.19$ 
C)  $0,2.09,3.14,4.19$ 
D)  $0,1.05,3.14,5.24$

Question 9

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

Accepted Answer

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

Question 10

Use Sum and Difference Formulas for Cosines and Sines
-$\sin 255 ^ { \circ }$

Accepted Answer

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

Question 11

Verify the identity.
-$$\tan ^ { 2 } x ( 1 + \cos 2 x ) = 1 - \cos 2 x$$

Accepted Answer

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

Question 12

Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal.
-$\cos 2 x + \cos 6 x = 2 \cos 4 x \cos x$

Accepted Answer

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

Question 13

Complete the identity.
-$\tan x ( \cot x - \cos x ) =$ ?

Accepted Answer

A)  $1 - \sin x$ 
B)  $- \sec ^ { 2 } x$ 
C)  1 
D)  0 
A)  $1 - \sin x$ 
B)  $- \sec ^ { 2 } x$ 
C)  1 
D)  0

Question 14

Use Identities to Solve Trigonometric Equations
Solve the equation on the interval [0, 2π).
-$$\cos \left( x + \frac { \pi } { 4 } \right) - \cos \left( x - \frac { \pi } { 4 } \right) = 1$$

Accepted Answer

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

Question 15

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

Accepted Answer

A)  $\frac { 1 } { 2 } ( \sin 5 x - \sin 4 x )$ 
B)  $\frac { 1 } { 4 } ( \cos 10 x - \sin 8 x )$ 
C)  $\frac { 1 } { 4 } \sin ( \cos 9 x )$ 
D)  $\frac { 1 } { 2 } ( \cos 5 x + \sin 4 x )$ 
A)  $\frac { 1 } { 2 } ( \sin 5 x - \sin 4 x )$ 
B)  $\frac { 1 } { 4 } ( \cos 10 x - \sin 8 x )$ 
C)  $\frac { 1 } { 4 } \sin ( \cos 9 x )$ 
D)  $\frac { 1 } { 2 } ( \cos 5 x + \sin 4 x )$

Question 16

Use The Half-Angle Formulas
-$\sin \frac { 3 \pi } { 8 }$

Accepted Answer

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

Question 17

Complete the identity.
-$$\tan \frac { \theta } { 2 } + \cot \frac { \theta } { 2 } = ?$$

Accepted Answer

A)  $2 \csc \theta$ 
B)  $- 2 \csc \theta$ 
C)  $- 2 \cot \theta$ 
D)  $2 \cot \theta$ 
A)  $2 \csc \theta$ 
B)  $- 2 \csc \theta$ 
C)  $- 2 \cot \theta$ 
D)  $2 \cot \theta$

Question 18

Use substitution to determine whether the given x-value is a solution of the equation.
Find All Solutions of a Trigonometric Equation
-$\tan x \sec x = - 2 \tan x$

Accepted Answer

A)  $x = \frac { 2 \pi } { 3 } + 2 n \pi$ or $x = \frac { 4 \pi } { 3 } + 2 n \pi$ or $x = n \pi$ 
B)  $x = \frac { 2 \pi } { 3 } + n \pi$ or $x = \frac { 4 \pi } { 3 } + n \pi$ or $x = n \pi$ 
C)  $x = \frac { \pi } { 3 } + 2 n \pi$ or $x = \frac { 5 \pi } { 3 } + 2 n \pi$ or $x = n \pi$ 
D)  $x = \frac { \pi } { 3 } + n \pi$ or $x = \frac { 5 \pi } { 3 } + n \pi$ or $x = n \pi$ 
A)  $x = \frac { 2 \pi } { 3 } + 2 n \pi$ or $x = \frac { 4 \pi } { 3 } + 2 n \pi$ or $x = n \pi$ 
B)  $x = \frac { 2 \pi } { 3 } + n \pi$ or $x = \frac { 4 \pi } { 3 } + n \pi$ or $x = n \pi$ 
C)  $x = \frac { \pi } { 3 } + 2 n \pi$ or $x = \frac { 5 \pi } { 3 } + 2 n \pi$ or $x = n \pi$ 
D)  $x = \frac { \pi } { 3 } + n \pi$ or $x = \frac { 5 \pi } { 3 } + n \pi$ or $x = n \pi$

Question 19

Verify the identity.
-$$\tan \left( x - \frac { \pi } { 4 } \right) = \frac { \tan x - 1 } { 1 + \tan x }$$

Accepted Answer

The answer of Verify the identity.
-\[\tan \left( x - \frac...

Question 20

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

Accepted Answer

A)  $\sec x - \tan x$ 
B)  $\sec x + \tan x$ 
C)  $- \sec x - \tan x$ 
D)  $\sec x - \tan x + 1$ 
A)  $\sec x - \tan x$ 
B)  $\sec x + \tan x$ 
C)  $- \sec x - \tan x$ 
D)  $\sec x - \tan x + 1$

Use the Power-Reducing Formulas - $\sin ^ { 4 } x$

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

Use Sum and Difference Formulas for Cosines and Sines - $\cos \left( 30 ^ { \circ } + 45 ^ { \circ } \right)$

Verify the identity. - $\cos \left( \frac { 3 \pi } { 2 } - \theta \right) = - \sin \theta$

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

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

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

Use Sum and Difference Formulas for Cosines and Sines - $\sin 255 ^ { \circ }$

Verify the identity. - $\tan ^ { 2 } x ( 1 + \cos 2 x ) = 1 - \cos 2 x$

Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal. - $\cos 2 x + \cos 6 x = 2 \cos 4 x \cos x$

Complete the identity. - $\tan x ( \cot x - \cos x ) =$ ?

Use Identities to Solve Trigonometric Equations Solve the equation on the interval [0, 2π). - $\cos \left( x + \frac { \pi } { 4 } \right) - \cos \left( x - \frac { \pi } { 4 } \right) = 1$

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

Use The Half-Angle Formulas - $\sin \frac { 3 \pi } { 8 }$

Complete the identity. - $\tan \frac { \theta } { 2 } + \cot \frac { \theta } { 2 } = ?$

Use substitution to determine whether the given x-value is a solution of the equation. Find All Solutions of a Trigonometric Equation - $\tan x \sec x = - 2 \tan x$

Verify the identity. - $\tan \left( x - \frac { \pi } { 4 } \right) = \frac { \tan x - 1 } { 1 + \tan x }$

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

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 the Power-Reducing Formulas - sin⁡4x\sin ^ { 4 } xsin4x

Complete the identity. - (sin⁡x+cos⁡x)21+2sin⁡xcos⁡x=?\frac { ( \sin x + \cos x ) ^ { 2 } } { 1 + 2 \sin x \cos x } = ?1+2sinxcosx(sinx+cosx)2​=?

Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal. - cos⁡10x−cos⁡4xsin⁡10x+sin⁡4x= ? \frac { \cos 10 x - \cos 4 x } { \sin 10 x + \sin 4 x } = \text { ? }sin10x+sin4xcos10x−cos4x​= ?

Use Sum and Difference Formulas for Cosines and Sines - cos⁡(30∘+45∘)\cos \left( 30 ^ { \circ } + 45 ^ { \circ } \right)cos(30∘+45∘)

Verify the identity. - cos⁡(3π2−θ)=−sin⁡θ\cos \left( \frac { 3 \pi } { 2 } - \theta \right) = - \sin \thetacos(23π​−θ)=−sinθ

Complete the identity. - 1+cos⁡2xsin⁡2x= ? \frac { 1 + \cos 2 x } { \sin 2 x } = \text { ? }sin2x1+cos2x​= ?

Use a calculator to solve the equation on the interval [0, 2π). Round the answer to two decimal places. - cos⁡2x+cos⁡x=0\cos 2 x + \cos x = 0cos2x+cosx=0

Complete the identity. - sin⁡2x+sin⁡2xcot⁡2x=\sin ^ { 2 } x + \sin ^ { 2 } x \cot ^ { 2 } x =sin2x+sin2xcot2x= ?

Use Sum and Difference Formulas for Cosines and Sines - sin⁡255∘\sin 255 ^ { \circ }sin255∘

Verify the identity. - tan⁡2x(1+cos⁡2x)=1−cos⁡2x\tan ^ { 2 } x ( 1 + \cos 2 x ) = 1 - \cos 2 xtan2x(1+cos2x)=1−cos2x

Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal. - cos⁡2x+cos⁡6x=2cos⁡4xcos⁡x\cos 2 x + \cos 6 x = 2 \cos 4 x \cos xcos2x+cos6x=2cos4xcosx

Complete the identity. - tan⁡x(cot⁡x−cos⁡x)=\tan x ( \cot x - \cos x ) =tanx(cotx−cosx)= ?

Use Identities to Solve Trigonometric Equations Solve the equation on the interval [0, 2π). - cos⁡(x+π4)−cos⁡(x−π4)=1\cos \left( x + \frac { \pi } { 4 } \right) - \cos \left( x - \frac { \pi } { 4 } \right) = 1cos(x+4π​)−cos(x−4π​)=1

Express the product as a sum or difference. - sin⁡x2cos⁡9x2\sin \frac { x } { 2 } \cos \frac { 9 x } { 2 }sin2x​cos29x​

Use The Half-Angle Formulas - sin⁡3π8\sin \frac { 3 \pi } { 8 }sin83π​

Complete the identity. - tan⁡θ2+cot⁡θ2=?\tan \frac { \theta } { 2 } + \cot \frac { \theta } { 2 } = ?tan2θ​+cot2θ​=?

Use substitution to determine whether the given x-value is a solution of the equation. Find All Solutions of a Trigonometric Equation - tan⁡xsec⁡x=−2tan⁡x\tan x \sec x = - 2 \tan xtanxsecx=−2tanx

Verify the identity. - tan⁡(x−π4)=tan⁡x−11+tan⁡x\tan \left( x - \frac { \pi } { 4 } \right) = \frac { \tan x - 1 } { 1 + \tan x }tan(x−4π​)=1+tanxtanx−1​

Complete the identity. - 1−sin⁡xcos⁡x=?\frac { 1 - \sin x } { \cos x } = ?cosx1−sinx​=?