Question 1

Solve the equation for exact solutions over the interval [0, 2 $[ 0,2 \pi )$
-$$\sin ^ { 2 } x - \cos ^ { 2 } x = 0$$

Accepted Answer

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

Question 2

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

Accepted Answer

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

Question 3

Give the exact value of the expression.
-$$\cos \left( \arcsin \frac { 5 } { 13 } + \arccos \frac { 3 } { 5 } \right)$$

Accepted Answer

A)  $\frac { 16 } { 65 }$ 
B)  $\frac { 72 } { 65 }$ 
C)  $\frac { 56 } { 65 }$ 
D)  $\frac { 48 } { 65 }$ 
A)  $\frac { 16 } { 65 }$ 
B)  $\frac { 72 } { 65 }$ 
C)  $\frac { 56 } { 65 }$ 
D)  $\frac { 48 } { 65 }$

Question 4

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-$$\frac { \tan \theta } { \sec \theta }$$

Accepted Answer

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

Question 5

Solve the equation for exact solutions.
-$\arcsin x + 2 \arctan x = \pi$

Accepted Answer

A)  $\left\{ - \frac { \sqrt { 3 } } { 2 } , \frac { \sqrt { 3 } } { 2 } \right\}$ 
B)  $\{ 0 \}$ 
C)  $\{ 1 \}$ 
D)  $\left\{ - \frac { \sqrt { 3 } } { 4 } , \frac { \sqrt { 3 } } { 4 } \right\}$ 
A)  $\left\{ - \frac { \sqrt { 3 } } { 2 } , \frac { \sqrt { 3 } } { 2 } \right\}$ 
B)  $\{ 0 \}$ 
C)  $\{ 1 \}$ 
D)  $\left\{ - \frac { \sqrt { 3 } } { 4 } , \frac { \sqrt { 3 } } { 4 } \right\}$

Question 6

Solve the equation for exact solutions.
-A rotating beacon is located a distance $\mathrm { d }$ from a long wall. The distance $\mathrm { d }$ is given by $\mathrm { d } = 2 \tan 2 \pi \mathrm { t }$
where $t$ is the time measured in seconds since the beacon started rotating. Solve the equation for $t$.

Accepted Answer

A)  $\mathrm { t } = \frac { 1 } { 2 \pi } \arctan \frac { \mathrm { d } } { 2 }$ 
B)  $\mathrm { t } = 2 \pi \tan \frac { \mathrm { d } } { 2 }$ 
C)  $\mathrm { t } = 2 \pi \arctan \frac { \mathrm { d } } { 2 }$ 
D)  $\mathrm { t } = \frac { 1 } { 2 \pi } \tan \frac { \mathrm { d } } { 2 }$ 
A)  $\mathrm { t } = \frac { 1 } { 2 \pi } \arctan \frac { \mathrm { d } } { 2 }$ 
B)  $\mathrm { t } = 2 \pi \tan \frac { \mathrm { d } } { 2 }$ 
C)  $\mathrm { t } = 2 \pi \arctan \frac { \mathrm { d } } { 2 }$ 
D)  $\mathrm { t } = \frac { 1 } { 2 \pi } \tan \frac { \mathrm { d } } { 2 }$

Question 7

Verify that each equation is an identity.
-$$\frac { \tan x + \cot x } { \tan x - \cot x } = \frac { 1 } { \sin ^ { 2 } x - \cos ^ { 2 } x }$$

Accepted Answer

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

Question 8

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

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 9

Use a sum or difference identity to find the exact value.
-$$\frac { \tan 35 ^ { \circ } + \tan 115 ^ { \circ } } { 1 - \tan 35 ^ { \circ } \tan 115 ^ { \circ } }$$

Accepted Answer

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

Question 10

Write the product as a sum or difference of trigonometric functions.
-$\cos 38 ^ { \circ } \sin 23 ^ { \circ }$

Accepted Answer

A)  $\frac { 1 } { 2 } \left( \cos 61 ^ { \circ } + \cos 15 ^ { \circ } \right)$ 
B)  $\frac { 1 } { 2 } \left( \cos 61 ^ { \circ } - \cos 15 ^ { \circ } \right)$ 
C)  $\frac { 1 } { 2 } \left( \sin 61 ^ { \circ } - \sin 15 ^ { \circ } \right)$ 
D)  $\frac { 1 } { 2 } \left( \sin 61 ^ { \circ } + \sin 15 ^ { \circ } \right)$ 
A)  $\frac { 1 } { 2 } \left( \cos 61 ^ { \circ } + \cos 15 ^ { \circ } \right)$ 
B)  $\frac { 1 } { 2 } \left( \cos 61 ^ { \circ } - \cos 15 ^ { \circ } \right)$ 
C)  $\frac { 1 } { 2 } \left( \sin 61 ^ { \circ } - \sin 15 ^ { \circ } \right)$ 
D)  $\frac { 1 } { 2 } \left( \sin 61 ^ { \circ } + \sin 15 ^ { \circ } \right)$

Question 11

Verify that each equation is an identity.
-$$\sin ^ { 3 } x \cos ^ { 2 } x = \sin x \left( \cos ^ { 2 } x - \cos ^ { 4 } x \right)$$

Accepted Answer

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

Question 12

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$$y = \cos ^ { - 1 } ( - 0.9397 )$$

Accepted Answer

A)  $2.7925484$ 
B)  $2.8285740$ 
C)  $- 0.3492063$ 
D)  $2.7412723$ 
A)  $2.7925484$ 
B)  $2.8285740$ 
C)  $- 0.3492063$ 
D)  $2.7412723$

Question 13

Use the fundamental identities to simplify the expression.
-$$\frac { \sec ^ { 2 } x } { \tan x } - \tan x$$

Accepted Answer

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

Question 14

Verify that the equation is an identity.
-A vibrating wire has a vertical displacement given by $y = 4 ( \cos 2 \pi x \cos b t + \sin 2 \pi x \sin b t )$. Assume $\mathrm { b }$ and $\mathrm { t }$ are constants. Write $\mathrm { y }$ as a cosine function of $\mathrm { x }$.

Accepted Answer

A)  $y = 8 \cos ( 2 \pi x + b t )$ 
B)  $y = 4 \cos ( 2 \pi x - b t )$ 
C)  $y = \cos ( 2 \pi x - b t )$ 
D)  $y = 4 \cos ( 2 \pi x + b t )$ 
A)  $y = 8 \cos ( 2 \pi x + b t )$ 
B)  $y = 4 \cos ( 2 \pi x - b t )$ 
C)  $y = \cos ( 2 \pi x - b t )$ 
D)  $y = 4 \cos ( 2 \pi x + b t )$

Question 15

Use the fundamental identities to find the value of the trigonometric function.
-Find $\csc \theta$ if $\cot \theta = - \sqrt { 15 }$ and $\theta$ is in quadrant II.

Accepted Answer

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

Question 16

Use the cofunction identities to find an angle $$\theta$$ that makes the statement true.
-$$\sec \left( 6 \theta + 17 ^ { \circ } \right) = \csc \left( 2 \theta - 7 ^ { \circ } \right)$$

Accepted Answer

A)  $\theta = \frac { 17 ^ { \circ } } { 7 }$ 
B)  $\theta = 10 ^ { \circ }$ 
C)  $\theta = \frac { 83 ^ { \circ } } { 7 }$ 
D)  $\theta = 40 ^ { \circ }$ 
A)  $\theta = \frac { 17 ^ { \circ } } { 7 }$ 
B)  $\theta = 10 ^ { \circ }$ 
C)  $\theta = \frac { 83 ^ { \circ } } { 7 }$ 
D)  $\theta = 40 ^ { \circ }$

Question 17

Solve the equation for exact solutions.
-In the study of alternating electric current, capacitive voltage is given by $$V _ { C } = - \frac { i } { 2 \pi f } \cos 2 \pi f t$$ where f is the number of cycles per second, i is the maximum current, and t is the time in seconds. Solve the equation for t.

Accepted Answer

A)  $t = 2 \pi f \cos \left( - \frac { 2 \pi f V c } { i } \right)$ 
B)  $t = 2 \pi f \arccos \left( - \frac { 2 \pi \mathrm { fVc } } { \mathrm { i } } \right)$ 
C)  $\mathrm { t } = \frac { 1 } { 2 \pi \mathrm { f } } \arccos \left( \frac { 2 \pi \mathrm { fVc } } { \mathrm { i } } \right)$ 
D)  $t = \frac { 1 } { 2 \pi f } \arccos \left( - \frac { 2 \pi f V c } { i } \right)$ 
A)  $t = 2 \pi f \cos \left( - \frac { 2 \pi f V c } { i } \right)$ 
B)  $t = 2 \pi f \arccos \left( - \frac { 2 \pi \mathrm { fVc } } { \mathrm { i } } \right)$ 
C)  $\mathrm { t } = \frac { 1 } { 2 \pi \mathrm { f } } \arccos \left( \frac { 2 \pi \mathrm { fVc } } { \mathrm { i } } \right)$ 
D)  $t = \frac { 1 } { 2 \pi f } \arccos \left( - \frac { 2 \pi f V c } { i } \right)$

Question 18

Solve the equation (x in radians and $$\boldsymbol { \theta }$$ in degrees) for all exact solutions where appropriate. Round approximate answers
in radians to four decimal places and approximate answers in degrees to the nearest tenth.
-$4 \sin ^ { 2 } x - 1 = 0$

Accepted Answer

A)  $\left\{ \frac { \pi } { 3 } + n \pi , \frac { 5 \pi } { 6 } + n \pi \right\}$ 
B)  $\left\{ \frac { \pi } { 6 } + n \pi , \frac { 5 \pi } { 6 } + n \pi \right\}$ 
C)  $\left\{ \frac { \pi } { 6 } + n \pi , \frac { \pi } { 2 } + 2 n \pi \right\}$ 
D)  $\left\{ \frac { \pi } { 3 } + n \pi \right\}$ 
A)  $\left\{ \frac { \pi } { 3 } + n \pi , \frac { 5 \pi } { 6 } + n \pi \right\}$ 
B)  $\left\{ \frac { \pi } { 6 } + n \pi , \frac { 5 \pi } { 6 } + n \pi \right\}$ 
C)  $\left\{ \frac { \pi } { 6 } + n \pi , \frac { \pi } { 2 } + 2 n \pi \right\}$ 
D)  $\left\{ \frac { \pi } { 3 } + n \pi \right\}$

Question 19

Find the exact value by using a half-angle identity.
-$\tan 165 ^ { \circ }$

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 20

Write the following as an algebraic expression in u, u > 0.
-$\sin \left( \arctan \frac { \mathrm { u } } { \sqrt { 3 } } \right)$

Accepted Answer

A)  $\frac { u \sqrt { u ^ { 2 } - 3 } } { u ^ { 2 } - 3 }$ 
B)  $\mathrm { u } \sqrt { \mathrm { u } ^ { 2 } + 3 }$ 
C)  $\frac { \sqrt { u ^ { 2 } + 3 } } { u ^ { 2 } + 3 }$ 
D)  $\frac { u \sqrt { u ^ { 2 } + 3 } } { u ^ { 2 } + 3 }$ 
A)  $\frac { u \sqrt { u ^ { 2 } - 3 } } { u ^ { 2 } - 3 }$ 
B)  $\mathrm { u } \sqrt { \mathrm { u } ^ { 2 } + 3 }$ 
C)  $\frac { \sqrt { u ^ { 2 } + 3 } } { u ^ { 2 } + 3 }$ 
D)  $\frac { u \sqrt { u ^ { 2 } + 3 } } { u ^ { 2 } + 3 }$

Solve the equation for exact solutions over the interval [0, 2 $[ 0,2 \pi )$ - $\sin ^ { 2 } x - \cos ^ { 2 } x = 0$

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

Give the exact value of the expression. - $\cos \left( \arcsin \frac { 5 } { 13 } + \arccos \frac { 3 } { 5 } \right)$

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\frac { \tan \theta } { \sec \theta }$

Solve the equation for exact solutions. - $\arcsin x + 2 \arctan x = \pi$

Verify that each equation is an identity. - $\frac { \tan x + \cot x } { \tan x - \cot x } = \frac { 1 } { \sin ^ { 2 } x - \cos ^ { 2 } x }$

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

Use a sum or difference identity to find the exact value. - $\frac { \tan 35 ^ { \circ } + \tan 115 ^ { \circ } } { 1 - \tan 35 ^ { \circ } \tan 115 ^ { \circ } }$

Write the product as a sum or difference of trigonometric functions. - $\cos 38 ^ { \circ } \sin 23 ^ { \circ }$

Verify that each equation is an identity. - $\sin ^ { 3 } x \cos ^ { 2 } x = \sin x \left( \cos ^ { 2 } x - \cos ^ { 4 } x \right)$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $y = \cos ^ { - 1 } ( - 0.9397 )$

Use the fundamental identities to simplify the expression. - $\frac { \sec ^ { 2 } x } { \tan x } - \tan x$

Verify that the equation is an identity. -A vibrating wire has a vertical displacement given by $y = 4 ( \cos 2 \pi x \cos b t + \sin 2 \pi x \sin b t )$ . Assume $\mathrm { b }$ and $\mathrm { t }$ are constants. Write $\mathrm { y }$ as a cosine function of $\mathrm { x }$ .

Use the fundamental identities to find the value of the trigonometric function. -Find $\csc \theta$ if $\cot \theta = - \sqrt { 15 }$ and $\theta$ is in quadrant II.

Use the cofunction identities to find an angle $\theta$ that makes the statement true. - $\sec \left( 6 \theta + 17 ^ { \circ } \right) = \csc \left( 2 \theta - 7 ^ { \circ } \right)$

Solve the equation for exact solutions. -In the study of alternating electric current, capacitive voltage is given by $V _ { C } = - \frac { i } { 2 \pi f } \cos 2 \pi f t$ where f is the number of cycles per second, i is the maximum current, and t is the time in seconds. Solve the equation for t.

Solve the equation (x in radians and $\boldsymbol { \theta }$ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. - $4 \sin ^ { 2 } x - 1 = 0$

Find the exact value by using a half-angle identity. - $\tan 165 ^ { \circ }$

Write the following as an algebraic expression in u, u > 0. - $\sin \left( \arctan \frac { \mathrm { u } } { \sqrt { 3 } } \right)$

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Filters

Exam 8: Trigonometric Identities and Equations

Solve the equation for exact solutions over the interval [0, 2 [0,2π)[ 0,2 \pi )[0,2π) - sin⁡2x−cos⁡2x=0\sin ^ { 2 } x - \cos ^ { 2 } x = 0sin2x−cos2x=0

Give the degree measure of θ\thetaθ - θ=cos⁡−1(22)\theta = \cos ^ { - 1 } \left( \frac { \sqrt { 2 } } { 2 } \right)θ=cos−1(22​​)

Give the exact value of the expression. - cos⁡(arcsin⁡513+arccos⁡35)\cos \left( \arcsin \frac { 5 } { 13 } + \arccos \frac { 3 } { 5 } \right)cos(arcsin135​+arccos53​)

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - tan⁡θsec⁡θ\frac { \tan \theta } { \sec \theta }secθtanθ​

Solve the equation for exact solutions. - arcsin⁡x+2arctan⁡x=π\arcsin x + 2 \arctan x = \piarcsinx+2arctanx=π

Verify that each equation is an identity. - tan⁡x+cot⁡xtan⁡x−cot⁡x=1sin⁡2x−cos⁡2x\frac { \tan x + \cot x } { \tan x - \cot x } = \frac { 1 } { \sin ^ { 2 } x - \cos ^ { 2 } x }tanx−cotxtanx+cotx​=sin2x−cos2x1​

Use a sum or difference identity to find the exact value. - tan⁡11π12\tan \frac { 11 \pi } { 12 }tan1211π​

Use a sum or difference identity to find the exact value. - tan⁡35∘+tan⁡115∘1−tan⁡35∘tan⁡115∘\frac { \tan 35 ^ { \circ } + \tan 115 ^ { \circ } } { 1 - \tan 35 ^ { \circ } \tan 115 ^ { \circ } }1−tan35∘tan115∘tan35∘+tan115∘​

Write the product as a sum or difference of trigonometric functions. - cos⁡38∘sin⁡23∘\cos 38 ^ { \circ } \sin 23 ^ { \circ }cos38∘sin23∘

Verify that each equation is an identity. - sin⁡3xcos⁡2x=sin⁡x(cos⁡2x−cos⁡4x)\sin ^ { 3 } x \cos ^ { 2 } x = \sin x \left( \cos ^ { 2 } x - \cos ^ { 4 } x \right)sin3xcos2x=sinx(cos2x−cos4x)

Use a calculator to give the real number value. Round the answer to 7 decimal places. - y=cos⁡−1(−0.9397)y = \cos ^ { - 1 } ( - 0.9397 )y=cos−1(−0.9397)

Use the fundamental identities to simplify the expression. - sec⁡2xtan⁡x−tan⁡x\frac { \sec ^ { 2 } x } { \tan x } - \tan xtanxsec2x​−tanx

Use the fundamental identities to find the value of the trigonometric function. -Find csc⁡θ\csc \thetacscθ if cot⁡θ=−15\cot \theta = - \sqrt { 15 }cotθ=−15​ and θ\thetaθ is in quadrant II.

Use the cofunction identities to find an angle θ\thetaθ that makes the statement true. - sec⁡(6θ+17∘)=csc⁡(2θ−7∘)\sec \left( 6 \theta + 17 ^ { \circ } \right) = \csc \left( 2 \theta - 7 ^ { \circ } \right)sec(6θ+17∘)=csc(2θ−7∘)

Solve the equation (x in radians and θ\boldsymbol { \theta }θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. - 4sin⁡2x−1=04 \sin ^ { 2 } x - 1 = 04sin2x−1=0

Find the exact value by using a half-angle identity. - tan⁡165∘\tan 165 ^ { \circ }tan165∘

Write the following as an algebraic expression in u, u > 0. - sin⁡(arctan⁡u3)\sin \left( \arctan \frac { \mathrm { u } } { \sqrt { 3 } } \right)sin(arctan3​u​)