Question 1

Solve the equation for exact solutions.
-$$\cos ^ { - 1 } x = \sin ^ { - 1 } \frac { 4 } { 5 }$$

Accepted Answer

A)  $\left\{ \frac { 3 } { 4 } \right\}$ 
B)  $\varnothing$ 
C)  $\left\{ \frac { 3 } { 5 } \right\}$ 
D)  $\{ 0 \}$ 
A)  $\left\{ \frac { 3 } { 4 } \right\}$ 
B)  $\varnothing$ 
C)  $\left\{ \frac { 3 } { 5 } \right\}$ 
D)  $\{ 0 \}$

Question 2

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary.
-$3 \sin ^ { 2 } \theta - \sin \theta - 4 = 0$

Accepted Answer

A)  $\left\{ 180 ^ { \circ } \right\}$ 
B)  $\left\{ 270 ^ { \circ } \right\}$ 
C)  $\left\{ 0 ^ { \circ } \right\}$ 
D)  $\left\{ 90 ^ { \circ } \right\}$ 
A)  $\left\{ 180 ^ { \circ } \right\}$ 
B)  $\left\{ 270 ^ { \circ } \right\}$ 
C)  $\left\{ 0 ^ { \circ } \right\}$ 
D)  $\left\{ 90 ^ { \circ } \right\}$

Question 3

Write in terms of the cofunction of a complementary angle.
-$\sec 72 ^ { \circ } 18 ^ { \prime }$

Accepted Answer

A)  $\csc 17 ^ { \circ } 42 ^ { \prime }$ 
B)  $\csc 18 ^ { \circ } 18 ^ { \prime }$ 
C)  $\cos 17 ^ { \circ } 42 ^ { \prime }$ 
D)  $\cos 107 ^ { \circ } 42 ^ { \prime }$ 
A)  $\csc 17 ^ { \circ } 42 ^ { \prime }$ 
B)  $\csc 18 ^ { \circ } 18 ^ { \prime }$ 
C)  $\cos 17 ^ { \circ } 42 ^ { \prime }$ 
D)  $\cos 107 ^ { \circ } 42 ^ { \prime }$

Question 4

Use the fundamental identities to simplify the expression.
-$$1 - \cos ^ { 2 } \theta$$

Accepted Answer

A)  $\sin ^ { 2 } \theta$ 
B)  $\sin \theta$ 
C)  $- \sin ^ { 2 } \theta$ 
D)  $\tan ^ { 2 } \theta$ 
A)  $\sin ^ { 2 } \theta$ 
B)  $\sin \theta$ 
C)  $- \sin ^ { 2 } \theta$ 
D)  $\tan ^ { 2 } \theta$

Question 5

Decide whether the expression is or is not an identity.
-$$\sqrt { \sin ^ { 2 } x } = \sin x$$

Accepted Answer

A)  Identity 
B)  Not an identity 
A)  Identity 
B)  Not an identity

Question 6

Verify that each equation is an identity.
-$$\frac { \sec \theta - 1 } { \tan \theta } = \frac { \tan \theta } { \sec \theta + 1 }$$

Accepted Answer

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

Question 7

Find the exact value of the expression using the provided information.
-Find $\cos ( s - t )$ given that $\cos s = - \frac { 1 } { 6 }$, with $s$ in quadrant III, and $\cos t = - \frac { 3 } { 5 }$, with $t$ in quadrant III.

Accepted Answer

A)  $\frac { 3 - 4 \sqrt { 37 } } { 30 }$ 
B)  $\frac { 3 - 4 \sqrt { 35 } } { 30 }$ 
C)  $\frac { 3 + 4 \sqrt { 35 } } { 30 }$ 
D)  $\frac { - 3 + 4 \sqrt { 35 } } { 30 }$ 
A)  $\frac { 3 - 4 \sqrt { 37 } } { 30 }$ 
B)  $\frac { 3 - 4 \sqrt { 35 } } { 30 }$ 
C)  $\frac { 3 + 4 \sqrt { 35 } } { 30 }$ 
D)  $\frac { - 3 + 4 \sqrt { 35 } } { 30 }$

Question 8

Solve the equation for exact solutions.
-$\arccos x + \arccos 2 x = \arccos \frac { 1 } { 2 }$

Accepted Answer

A)  $\left\{ - \frac { 3 } { 4 } , \frac { 3 } { 4 } \right\}$ 
B)  $\left\{ \frac { 1 } { 2 } \right\}$ 
C)  $\{ 2 \}$ 
D)  $\{ - 5 + 3 \sqrt { 2 } \}$ 
A)  $\left\{ - \frac { 3 } { 4 } , \frac { 3 } { 4 } \right\}$ 
B)  $\left\{ \frac { 1 } { 2 } \right\}$ 
C)  $\{ 2 \}$ 
D)  $\{ - 5 + 3 \sqrt { 2 } \}$

Question 9

Rewrite the following as a product of trigonometric functions.
-$$\cos \frac { \pi } { 11 } - \cos \frac { \pi } { 8 }$$

Accepted Answer

A)  $- 2 \sin \frac { 19 \pi } { 176 } \sin \left( \frac { - 3 \pi } { 176 } \right)$ 
B)  $- 2 \sin \frac { 19 \pi } { 176 } \cos \left( \frac { - 3 \pi } { 176 } \right)$ 
C)  $- 2 \cos \frac { 19 \pi } { 176 } \sin \left( \frac { - 3 \pi } { 176 } \right)$ 
D)  $- 2 \cos \frac { 19 \pi } { 176 } \cos \left( \frac { - 3 \pi } { 176 } \right)$ 
A)  $- 2 \sin \frac { 19 \pi } { 176 } \sin \left( \frac { - 3 \pi } { 176 } \right)$ 
B)  $- 2 \sin \frac { 19 \pi } { 176 } \cos \left( \frac { - 3 \pi } { 176 } \right)$ 
C)  $- 2 \cos \frac { 19 \pi } { 176 } \sin \left( \frac { - 3 \pi } { 176 } \right)$ 
D)  $- 2 \cos \frac { 19 \pi } { 176 } \cos \left( \frac { - 3 \pi } { 176 } \right)$

Question 10

Solve the problem.
-The range $\mathrm { r }$ of a projectile is given by
$$\mathrm { r } = \frac { 1 } { 32 } \mathrm { v } ^ { 2 } \sin 2 \theta \text {, }$$
where $\mathrm { v }$ is the initial velocity and $\theta$ is the angle of elevation. If $\mathrm { r }$ is to be $8000 \mathrm { ft }$ and $\mathrm { v } = 3000 \mathrm { ft } / \mathrm { sec }$, what must the angle of elevation be? Give your answer in degrees to the nearest hundredth.

Accepted Answer

A)  $1.14 ^ { \circ }$ 
B)  $1.63 ^ { \circ }$ 
C)  $89.19 ^ { \circ }$ 
D)  $0.81 ^ { \circ }$ 
A)  $1.14 ^ { \circ }$ 
B)  $1.63 ^ { \circ }$ 
C)  $89.19 ^ { \circ }$ 
D)  $0.81 ^ { \circ }$

Question 11

Write the product as a sum or difference of trigonometric functions.
-$$- 8 \cos ( - \theta ) \sin ( - 5 \theta )$$

Accepted Answer

A)  $- 4 ( \cos 6 \theta - \cos 4 \theta )$ 
B)  $4 ( \sin 6 \theta - \sin 4 \theta )$ 
C)  $- 8 ( \cos 6 \theta + \cos 4 \theta )$ 
D)  $4 ( \sin 6 \theta + \sin 4 \theta )$ 
A)  $- 4 ( \cos 6 \theta - \cos 4 \theta )$ 
B)  $4 ( \sin 6 \theta - \sin 4 \theta )$ 
C)  $- 8 ( \cos 6 \theta + \cos 4 \theta )$ 
D)  $4 ( \sin 6 \theta + \sin 4 \theta )$

Question 12

Complete the sentence so the result is an identity. Let x be any real number.
-

Accepted Answer

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

Question 13

Solve the equation for solutions over the interval [0, 2π). Write solutions as exact values or to four decimal places, as
appropriate.
-$$\sin \frac { x } { 2 } + \cos \frac { x } { 2 } = 0$$

Accepted Answer

A)  $\left\{ \frac { 3 \pi } { 2 } \right\}$ 
B)  $\{ 0 , \pi \}$ 
C)  $\left\{ \frac { 3 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
D)  $\left\{ \frac { \pi } { 2 } \right\}$ 
A)  $\left\{ \frac { 3 \pi } { 2 } \right\}$ 
B)  $\{ 0 , \pi \}$ 
C)  $\left\{ \frac { 3 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
D)  $\left\{ \frac { \pi } { 2 } \right\}$

Question 14

Use Identities to find the exact value.
-$\cos 36 ^ { \circ } \cos 24 ^ { \circ } - \sin 36 ^ { \circ } \sin 24 ^ { \circ }$

Accepted Answer

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

Question 15

Use a sum or difference identity to find the exact value.
-$\sin \frac { 11 \pi } { 12 }$

Accepted Answer

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

Question 16

Give the exact value of the expression.
-$\cos \left( \arcsin \frac { 1 } { 4 } \right)$

Accepted Answer

A)  $\frac { \sqrt { 15 } } { 4 }$ 
B)  $\frac { 2 \sqrt { 15 } } { 15 }$ 
C)  $\frac { \sqrt { 15 } } { 2 }$ 
D)  $\frac { 4 \sqrt { 15 } } { 15 }$ 
A)  $\frac { \sqrt { 15 } } { 4 }$ 
B)  $\frac { 2 \sqrt { 15 } } { 15 }$ 
C)  $\frac { \sqrt { 15 } } { 2 }$ 
D)  $\frac { 4 \sqrt { 15 } } { 15 }$

Question 17

Give the degree measure of θ.
-$$\theta = \cos ^ { - 1 } \left( \frac { \sqrt { 3 } } { 2 } \right)$$

Accepted Answer

A)  $30 ^ { \circ }$ 
B)  $315 ^ { \circ }$ 
C)  $45 ^ { \circ }$ 
D)  $330 ^ { \circ }$ 
A)  $30 ^ { \circ }$ 
B)  $315 ^ { \circ }$ 
C)  $45 ^ { \circ }$ 
D)  $330 ^ { \circ }$

Question 18

Perform the transformation.
-Write $\tan x$ in terms of $\sin x$.

Accepted Answer

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

Question 19

Use a sum or difference identity to find the exact value.
-$\sin 10 ^ { \circ } \cos 110 ^ { \circ } + \cos 10 ^ { \circ } \sin 110 ^ { \circ }$

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $- \frac { \sqrt { 3 } } { 2 }$ 
C)  $- \frac { 1 } { 2 }$ 
D)  $\frac { 1 } { 6 }$ 
A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $- \frac { \sqrt { 3 } } { 2 }$ 
C)  $- \frac { 1 } { 2 }$ 
D)  $\frac { 1 } { 6 }$

Question 20

Solve the problem.
-Let $( a , b )$ and (c, d) be two points in the first quadrant, and let $\theta$ be the angle between the line segment connecting $( a , b )$ with the origin and the line segment connecting $( c , d )$ with the origin. It can be shown that
$\cos \theta = \frac { a c + b d } { \sqrt { \left( a ^ { 2 } + b ^ { 2 } \right) \left( c ^ { 2 } + d ^ { 2 } \right) } }$
Find $\theta$ if $a = 6 , b = 8 , c = 8$, and $d = 2$. Give your answer in degrees rounded to the nearest hundredth.

Accepted Answer

A)  $67.17 ^ { \circ }$ 
B)  $78.81 ^ { \circ }$ 
C)  $39.09 ^ { \circ }$ 
D)  $54.40 ^ { \circ }$ 
A)  $67.17 ^ { \circ }$ 
B)  $78.81 ^ { \circ }$ 
C)  $39.09 ^ { \circ }$ 
D)  $54.40 ^ { \circ }$

Solve the equation for exact solutions. - $\cos ^ { - 1 } x = \sin ^ { - 1 } \frac { 4 } { 5 }$

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary. - $3 \sin ^ { 2 } \theta - \sin \theta - 4 = 0$

Write in terms of the cofunction of a complementary angle. - $\sec 72 ^ { \circ } 18 ^ { \prime }$

Use the fundamental identities to simplify the expression. - $1 - \cos ^ { 2 } \theta$

Decide whether the expression is or is not an identity. - $\sqrt { \sin ^ { 2 } x } = \sin x$

Verify that each equation is an identity. - $\frac { \sec \theta - 1 } { \tan \theta } = \frac { \tan \theta } { \sec \theta + 1 }$

Find the exact value of the expression using the provided information. -Find $\cos ( s - t )$ given that $\cos s = - \frac { 1 } { 6 }$ , with $s$ in quadrant III, and $\cos t = - \frac { 3 } { 5 }$ , with $t$ in quadrant III.

Solve the equation for exact solutions. - $\arccos x + \arccos 2 x = \arccos \frac { 1 } { 2 }$

Rewrite the following as a product of trigonometric functions. - $\cos \frac { \pi } { 11 } - \cos \frac { \pi } { 8 }$

Write the product as a sum or difference of trigonometric functions. - $- 8 \cos ( - \theta ) \sin ( - 5 \theta )$

Complete the sentence so the result is an identity. Let x be any real number. -

Solve the equation for solutions over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate. - $\sin \frac { x } { 2 } + \cos \frac { x } { 2 } = 0$

Use Identities to find the exact value. - $\cos 36 ^ { \circ } \cos 24 ^ { \circ } - \sin 36 ^ { \circ } \sin 24 ^ { \circ }$

Use a sum or difference identity to find the exact value. - $\sin \frac { 11 \pi } { 12 }$

Give the exact value of the expression. - $\cos \left( \arcsin \frac { 1 } { 4 } \right)$

Give the degree measure of θ. - $\theta = \cos ^ { - 1 } \left( \frac { \sqrt { 3 } } { 2 } \right)$

Perform the transformation. -Write $\tan x$ in terms of $\sin x$ .

Use a sum or difference identity to find the exact value. - $\sin 10 ^ { \circ } \cos 110 ^ { \circ } + \cos 10 ^ { \circ } \sin 110 ^ { \circ }$

Equations and Inequalities

Graphs and Functions

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Exam 7: Trigonometric Identities and Equations

Solve the equation for exact solutions. - cos⁡−1x=sin⁡−145\cos ^ { - 1 } x = \sin ^ { - 1 } \frac { 4 } { 5 }cos−1x=sin−154​

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary. - 3sin⁡2θ−sin⁡θ−4=03 \sin ^ { 2 } \theta - \sin \theta - 4 = 03sin2θ−sinθ−4=0

Write in terms of the cofunction of a complementary angle. - sec⁡72∘18′\sec 72 ^ { \circ } 18 ^ { \prime }sec72∘18′

Use the fundamental identities to simplify the expression. - 1−cos⁡2θ1 - \cos ^ { 2 } \theta1−cos2θ

Decide whether the expression is or is not an identity. - sin⁡2x=sin⁡x\sqrt { \sin ^ { 2 } x } = \sin xsin2x​=sinx

Verify that each equation is an identity. - sec⁡θ−1tan⁡θ=tan⁡θsec⁡θ+1\frac { \sec \theta - 1 } { \tan \theta } = \frac { \tan \theta } { \sec \theta + 1 }tanθsecθ−1​=secθ+1tanθ​

Find the exact value of the expression using the provided information. -Find cos⁡(s−t)\cos ( s - t )cos(s−t) given that cos⁡s=−16\cos s = - \frac { 1 } { 6 }coss=−61​ , with sss in quadrant III, and cos⁡t=−35\cos t = - \frac { 3 } { 5 }cost=−53​ , with ttt in quadrant III.

Solve the equation for exact solutions. - arccos⁡x+arccos⁡2x=arccos⁡12\arccos x + \arccos 2 x = \arccos \frac { 1 } { 2 }arccosx+arccos2x=arccos21​

Rewrite the following as a product of trigonometric functions. - cos⁡π11−cos⁡π8\cos \frac { \pi } { 11 } - \cos \frac { \pi } { 8 }cos11π​−cos8π​

Write the product as a sum or difference of trigonometric functions. - −8cos⁡(−θ)sin⁡(−5θ)- 8 \cos ( - \theta ) \sin ( - 5 \theta )−8cos(−θ)sin(−5θ)