Question 1

Use identities to find the indicated value for each angle measure.
-$2 \cos ^ { 2 } 22.5 ^ { \circ } - 1$

Accepted Answer

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

Question 2

Complete the sentence so the result is an identity. Let x be any real number.
-$\tan \theta = \frac { 7 } { 24 } , \quad \sin \theta < 0 \quad$ Find $\cos ( 2 \theta )$

Accepted Answer

A)  $\frac { 527 } { 625 }$ 
B)  $\frac { 336 } { 625 }$ 
C)  $- \frac { 527 } { 625 }$ 
D)  $- \frac { 336 } { 625 }$ 
A)  $\frac { 527 } { 625 }$ 
B)  $\frac { 336 } { 625 }$ 
C)  $- \frac { 527 } { 625 }$ 
D)  $- \frac { 336 } { 625 }$

Question 3

Find the exact value of the expression using the provided information.
-Find $\sin \theta$ if $\sec \theta = - \frac { 8 } { 5 }$ and $\tan \theta < 0$.

Accepted Answer

A)  $- \frac { \sqrt { 39 } } { 8 }$ 
B)  $\frac { \sqrt { 39 } } { 8 }$ 
C)  $- \frac { 5 } { 8 }$ 
D)  $\frac { 39 } { 64 }$ 
A)  $- \frac { \sqrt { 39 } } { 8 }$ 
B)  $\frac { \sqrt { 39 } } { 8 }$ 
C)  $- \frac { 5 } { 8 }$ 
D)  $\frac { 39 } { 64 }$

Question 4

Factor the trigonometric expression and simplify.
-$\sec ^ { 4 } x - 2 \sec ^ { 2 } x \tan ^ { 2 } x + \tan ^ { 4 } x$

Accepted Answer

A)  1 
B)  $\sec ^ { 2 } x \left( 1 + \tan ^ { 2 } x \right)$ 
C)  2 
D)  $\sec ^ { 2 } x + \tan ^ { 2 } x$ 
A)  1 
B)  $\sec ^ { 2 } x \left( 1 + \tan ^ { 2 } x \right)$ 
C)  2 
D)  $\sec ^ { 2 } x + \tan ^ { 2 } x$

Question 5

Use a sum or difference identity to find the exact value.
-

Accepted Answer

A)  $f ( - x ) = f ( x )$ 
B)  $f ( - x ) = - f ( x )$ 
A)  $f ( - x ) = f ( x )$ 
B)  $f ( - x ) = - f ( x )$

Question 6

Use the fundamental identities to simplify the expression.
-sin 13° - sin 44°

Accepted Answer

A) 2 cos 28.5° sin(-15.5°) 
B) 2 cos 28.5° cos(-15.5°) 
C) 2 sin 28.5° cos(-15.5°) 
D) 2 sin 28.5° sin(-15.5°) 
A) 2 cos 28.5° sin(-15.5°) 
B) 2 cos 28.5° cos(-15.5°) 
C) 2 sin 28.5° cos(-15.5°) 
D) 2 sin 28.5° sin(-15.5°)

Question 7

Using a sum or difference identity, write the following as an expression involving functions of x.
-Find $\cos \theta$ if $\sin \theta = - \frac { 5 } { 13 }$ and $\theta$ is in quadrant III.

Accepted Answer

A)  $- \frac { 5 } { 12 }$ 
B)  $- \frac { 12 } { 13 }$ 
C)  $\frac { 12 } { 5 }$ 
D)  $- \frac { 13 } { 5 }$ 
A)  $- \frac { 5 } { 12 }$ 
B)  $- \frac { 12 } { 13 }$ 
C)  $\frac { 12 } { 5 }$ 
D)  $- \frac { 13 } { 5 }$

Question 8

Use the fundamental identities to find the value of the trigonometric function.
-A wooden block sitting at rest on a flat, level board will just begin to slide when the board is tilted up to a critical angle of inclination, $\theta _ { \mathrm { S } }$. An experiment that determines this critical angle can be used to find the coefficient of static friction, $\mu _ { \mathrm { S } }$, for the block on the board by using the relation $\mu _ { \mathrm { S } } = \frac { \sin \theta _ { \mathrm { S } } } { \cos \theta _ { \mathrm { S } } }$. Rewrite this relation using a quotient identity.

Accepted Answer

A)  $\mu _ { \mathrm { S } } = \sec \theta _ { \mathrm { S } }$ 
B)  $\mu _ { \mathrm { s } } = \cot \theta _ { \mathrm { s } }$ 
C)  $\mu _ { \mathrm { s } } = \tan ^ { 2 } \theta _ { \mathrm { s } }$ 
D)  $\mu _ { \mathrm { s } } = \tan$ 
A)  $\mu _ { \mathrm { S } } = \sec \theta _ { \mathrm { S } }$ 
B)  $\mu _ { \mathrm { s } } = \cot \theta _ { \mathrm { s } }$ 
C)  $\mu _ { \mathrm { s } } = \tan ^ { 2 } \theta _ { \mathrm { s } }$ 
D)  $\mu _ { \mathrm { s } } = \tan$

Question 9

Use a sum or difference identity to find the exact value.
-Find $\cot \theta$ if $\tan \theta = \frac { \sqrt { 7 } } { 3 }$ and $\theta$ is in quadrant III.

Accepted Answer

A)  $\frac { 3 \sqrt { 7 } } { 7 }$ 
B)  $- \frac { 3 } { 2 }$ 
C)  $\frac { 5 } { 4 }$ 
D)  $- \frac { \sqrt { 7 } } { 9 }$ 
A)  $\frac { 3 \sqrt { 7 } } { 7 }$ 
B)  $- \frac { 3 } { 2 }$ 
C)  $\frac { 5 } { 4 }$ 
D)  $- \frac { \sqrt { 7 } } { 9 }$

Question 10

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.
-$$1 + --= \csc ^ { 2 } x$$

Accepted Answer

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

Question 11

Verify that each equation is an identity.
-$$\cot x \sec ^ { 4 } x = \cot x + 2 \tan x + \tan ^ { 3 } x$$

Accepted Answer

The answer of Verify that each equation is an identity.
-\[\cot...

Question 12

Use the cofunction identities to find an angle that makes the statement true.
-$$\frac { 1 + \tan ^ { 2 } x } { \sec x }$$

Accepted Answer

A)  1 
B)  $- \sec x$ 
C)  $\csc x$ 
D)  $\sec x$ 
A)  1 
B)  $- \sec x$ 
C)  $\csc x$ 
D)  $\sec x$

Question 13

Use the fundamental identities to simplify the expression.
-$\tan ^ { 2 } \theta \csc ^ { 2 } \theta$

Accepted Answer

A)  $\sin \theta$ 
B)  $\sec ^ { 2 } \theta$ 
C)  $\tan ^ { 2 } \theta$ 
D)  $\cos ^ { 3 } \theta$ 
A)  $\sin \theta$ 
B)  $\sec ^ { 2 } \theta$ 
C)  $\tan ^ { 2 } \theta$ 
D)  $\cos ^ { 3 } \theta$

Question 14

Use a sum or difference identity to find the exact value.
-$\sin 185 ^ { \circ } \cos 65 ^ { \circ } - \cos 185 ^ { \circ } \sin 65 ^ { \circ }$

Accepted Answer

A)  $- \frac { \sqrt { 3 } } { 2 }$ 
B)  $\frac { \sqrt { 3 } } { 2 }$ 
C)  $- \frac { 37 } { 12 }$ 
D)  $- \frac { 1 } { 2 }$ 
A)  $- \frac { \sqrt { 3 } } { 2 }$ 
B)  $\frac { \sqrt { 3 } } { 2 }$ 
C)  $- \frac { 37 } { 12 }$ 
D)  $- \frac { 1 } { 2 }$

Question 15

Use a sum or difference identity to find the exact value.
-Use the exact value $\cos 36 ^ { \circ } = \frac { \sqrt { 5 } + 1 } { 4 }$ and identities to find the exact value of $\sin 36 ^ { \circ }$.

Accepted Answer

A)  $\frac { ( \sqrt { 10 + 2 \sqrt { 5 } } ) ( 10 - 2 \sqrt { 5 } ) } { 5 }$ 
B)  $\sqrt { 5 } + 1$ 
C)  $\frac { \sqrt { 10 - 2 \sqrt { 5 } } } { 4 }$ 
D)  $\frac { \sqrt { 10 - 2 \sqrt { 5 } } } { 8 }$ 
A)  $\frac { ( \sqrt { 10 + 2 \sqrt { 5 } } ) ( 10 - 2 \sqrt { 5 } ) } { 5 }$ 
B)  $\sqrt { 5 } + 1$ 
C)  $\frac { \sqrt { 10 - 2 \sqrt { 5 } } } { 4 }$ 
D)  $\frac { \sqrt { 10 - 2 \sqrt { 5 } } } { 8 }$

Question 16

Solve the problem.
-In a particular situation, the pressure, P, exerted on a person's eardrum at a distance of 8 feet from the source is given by the function: $\mathrm { P } ( \mathrm { t } ) = \frac { \mathrm { a } } { 8 } \cos ( 2 \mathrm { n } \pi - \mathrm { ct } )$, where $\mathrm { n }$ is a positive integer. Use the difference identity for cosine to simplify $\mathrm { P }$ in this situation.

Accepted Answer

A)  $\mathrm { P } ( \mathrm { t } ) = \frac { \mathrm { a } } { 8 } \cos ( \mathrm { ct } )$ 
B)  $\mathrm { P } ( \mathrm { t } ) = \frac { 8 } { 9 } \cos ( \mathrm { ct } + 8 )$ 
C)  $\mathrm { P } ( \mathrm { t } ) = \frac { \mathrm { a } } { 8 } \cos ( \mathrm { ct } ) - \frac { \mathrm { a } } { 8 } \sin ( \mathrm { ct } )$ 
D)  $\mathrm { P } ( \mathrm { t } ) = \cos ( \mathrm { ct } )$ 
A)  $\mathrm { P } ( \mathrm { t } ) = \frac { \mathrm { a } } { 8 } \cos ( \mathrm { ct } )$ 
B)  $\mathrm { P } ( \mathrm { t } ) = \frac { 8 } { 9 } \cos ( \mathrm { ct } + 8 )$ 
C)  $\mathrm { P } ( \mathrm { t } ) = \frac { \mathrm { a } } { 8 } \cos ( \mathrm { ct } ) - \frac { \mathrm { a } } { 8 } \sin ( \mathrm { ct } )$ 
D)  $\mathrm { P } ( \mathrm { t } ) = \cos ( \mathrm { ct } )$

Question 17

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-tan 165°

Accepted Answer

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

Question 18

Use identities to find the indicated value for each angle measure.
-sin ϴ = cos (4ϴ + 60°)

Accepted Answer

A)  $\theta = 10 ^ { \circ }$ 
B)  $\theta = 6 ^ { \circ }$ 
C)  $\theta = 16 ^ { \circ }$ 
D)  $\theta = 75 ^ { \circ }$ 
A)  $\theta = 10 ^ { \circ }$ 
B)  $\theta = 6 ^ { \circ }$ 
C)  $\theta = 16 ^ { \circ }$ 
D)  $\theta = 75 ^ { \circ }$

Question 19

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-$\sin 115 ^ { \circ } \cos 55 ^ { \circ } - \cos 115 ^ { \circ } \sin 55 ^ { \circ }$

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 3 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 23 } { 12 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$ 
A)  $\frac { \sqrt { 3 } } { 3 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 23 } { 12 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$

Question 20

Use the cofunction identities to find an angle that makes the statement true.
-The output voltage of a generator is given by $\mathrm { V } = 158 \cos \left( 2 \pi \mathrm { ft } - \frac { \pi } { 6 } \right)$. Express the voltage as the sum of a sine and a cosine function.

Accepted Answer

A)  $V = 79 \sqrt { 3 } \cos 2 \pi \mathrm { ft } + 79 \sin 2 \pi \mathrm { ft }$ 
B)  $V = 79 \sqrt { 3 } \sin 2 \pi \mathrm { ft } + 79 \cos 2 \pi \mathrm { ft }$ 
C)  $V = 79 \cos 2 \pi \mathrm { ft } + 79 \sin 2 \pi \mathrm { ft }$ 
D)  $V = 79 \sqrt { 3 } \cos 2 \pi \mathrm { ft } - 79 \sin 2 \pi \mathrm { ft }$ 
A)  $V = 79 \sqrt { 3 } \cos 2 \pi \mathrm { ft } + 79 \sin 2 \pi \mathrm { ft }$ 
B)  $V = 79 \sqrt { 3 } \sin 2 \pi \mathrm { ft } + 79 \cos 2 \pi \mathrm { ft }$ 
C)  $V = 79 \cos 2 \pi \mathrm { ft } + 79 \sin 2 \pi \mathrm { ft }$ 
D)  $V = 79 \sqrt { 3 } \cos 2 \pi \mathrm { ft } - 79 \sin 2 \pi \mathrm { ft }$

Use identities to find the indicated value for each angle measure. - $2 \cos ^ { 2 } 22.5 ^ { \circ } - 1$

Complete the sentence so the result is an identity. Let x be any real number. - $\tan \theta = \frac { 7 } { 24 } , \quad \sin \theta < 0 \quad$ Find $\cos ( 2 \theta )$

Find the exact value of the expression using the provided information. -Find $\sin \theta$ if $\sec \theta = - \frac { 8 } { 5 }$ and $\tan \theta < 0$ .

Factor the trigonometric expression and simplify. - $\sec ^ { 4 } x - 2 \sec ^ { 2 } x \tan ^ { 2 } x + \tan ^ { 4 } x$

Use a sum or difference identity to find the exact value. -

Use the fundamental identities to simplify the expression. -sin 13° - sin 44°

Using a sum or difference identity, write the following as an expression involving functions of x. -Find $\cos \theta$ if $\sin \theta = - \frac { 5 } { 13 }$ and $\theta$ is in quadrant III.

Use a sum or difference identity to find the exact value. -Find $\cot \theta$ if $\tan \theta = \frac { \sqrt { 7 } } { 3 }$ and $\theta$ is in quadrant III.

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. - $1 + --= \csc ^ { 2 } x$

Verify that each equation is an identity. - $\cot x \sec ^ { 4 } x = \cot x + 2 \tan x + \tan ^ { 3 } x$

Use the cofunction identities to find an angle that makes the statement true. - $\frac { 1 + \tan ^ { 2 } x } { \sec x }$

Use the fundamental identities to simplify the expression. - $\tan ^ { 2 } \theta \csc ^ { 2 } \theta$

Use a sum or difference identity to find the exact value. - $\sin 185 ^ { \circ } \cos 65 ^ { \circ } - \cos 185 ^ { \circ } \sin 65 ^ { \circ }$

Use a sum or difference identity to find the exact value. -Use the exact value $\cos 36 ^ { \circ } = \frac { \sqrt { 5 } + 1 } { 4 }$ and identities to find the exact value of $\sin 36 ^ { \circ }$ .

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. -tan 165°

Use identities to find the indicated value for each angle measure. -sin ϴ = cos (4ϴ + 60°)

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\sin 115 ^ { \circ } \cos 55 ^ { \circ } - \cos 115 ^ { \circ } \sin 55 ^ { \circ }$

Use the cofunction identities to find an angle that makes the statement true. -The output voltage of a generator is given by $\mathrm { V } = 158 \cos \left( 2 \pi \mathrm { ft } - \frac { \pi } { 6 } \right)$ . Express the voltage as the sum of a sine and a cosine function.

Trigonometric Functions

Acute Angles and Right Triangles

Radian Measure and the Unit Circle

Graphs of the Circular Functions

Inverse Circular Functions and Trigonometric Equations

Applications of Trigonometry and Vectors

Complex Numbers, Polar Equations, and Parametric Equations

Filters

Exam 5: Trigonometric Identities

Use identities to find the indicated value for each angle measure. - 2cos⁡222.5∘−12 \cos ^ { 2 } 22.5 ^ { \circ } - 12cos222.5∘−1

Complete the sentence so the result is an identity. Let x be any real number. - tan⁡θ=724,sin⁡θ<0\tan \theta = \frac { 7 } { 24 } , \quad \sin \theta < 0 \quadtanθ=247​,sinθ<0 Find cos⁡(2θ)\cos ( 2 \theta )cos(2θ)

Find the exact value of the expression using the provided information. -Find sin⁡θ\sin \thetasinθ if sec⁡θ=−85\sec \theta = - \frac { 8 } { 5 }secθ=−58​ and tan⁡θ<0\tan \theta < 0tanθ<0 .

Factor the trigonometric expression and simplify. - sec⁡4x−2sec⁡2xtan⁡2x+tan⁡4x\sec ^ { 4 } x - 2 \sec ^ { 2 } x \tan ^ { 2 } x + \tan ^ { 4 } xsec4x−2sec2xtan2x+tan4x