Question 1

Use identities to write the expression as a single function of x or $$\theta$$
-$\tan \left( 30 ^ { \circ } + x \right)$

Accepted Answer

A)  $\frac { 1 + \sqrt { 3 } \tan x } { \sqrt { 3 } - \tan x }$ 
B)  $- \tan x$ 
C)  $\tan x$ 
D)  $\frac { \tan x - \sqrt { 3 } } { 1 + \sqrt { 3 } \tan x }$ 
A)  $\frac { 1 + \sqrt { 3 } \tan x } { \sqrt { 3 } - \tan x }$ 
B)  $- \tan x$ 
C)  $\tan x$ 
D)  $\frac { \tan x - \sqrt { 3 } } { 1 + \sqrt { 3 } \tan x }$

Question 2

Solve the equation for exact solutions.
-$6 \arcsin x = \pi$

Accepted Answer

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

Question 3

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

Accepted Answer

A)  $30 ^ { \circ }$ 
B)  $315 ^ { \circ }$ 
C)  $- 30 ^ { \circ }$ 
D)  $300 ^ { \circ }$ 
A)  $30 ^ { \circ }$ 
B)  $315 ^ { \circ }$ 
C)  $- 30 ^ { \circ }$ 
D)  $300 ^ { \circ }$

Question 4

Find the exact value of the expression using the provided information.
-Find $\tan ( \mathrm { s } + \mathrm { t } )$ given that $\sin \mathrm { s } = \frac { 1 } { 4 }$, with $\mathrm { s }$ in quadrant II, and $\sin \mathrm { t } = - \frac { 1 } { 2 }$, with $\mathrm { t }$ in quadrant IV.

Accepted Answer

A)  $\frac { 4 \sqrt { 3 } + \sqrt { 15 } } { - 11 }$ 
B)  $\frac { 1 + 3 \sqrt { 5 } } { 8 }$ 
C)  $\frac { 1 - 3 \sqrt { 5 } } { 8 }$ 
D)  $\frac { 4 \sqrt { 3 } - \sqrt { 15 } } { 11 }$ 
A)  $\frac { 4 \sqrt { 3 } + \sqrt { 15 } } { - 11 }$ 
B)  $\frac { 1 + 3 \sqrt { 5 } } { 8 }$ 
C)  $\frac { 1 - 3 \sqrt { 5 } } { 8 }$ 
D)  $\frac { 4 \sqrt { 3 } - \sqrt { 15 } } { 11 }$

Question 5

Complete the sentence so the result is an identity. Let x be any real number.
-$- 1 = \tan ^ { 2 } x$

Accepted Answer

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

Question 6

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as
appropriate.
-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 $\mathrm { f }$ is the number of cycles per second, $\mathrm { i }$ is the maximum current, and $\mathrm { t }$ is the time in seconds. Solve the equation for $\mathrm { t }$.

Accepted Answer

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

Question 7

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as
appropriate.
-In an electric circuit, the electromotive force is defined as $\mathrm { V } = \cos 2 \pi \mathrm { t }$, where $\mathrm { t }$ is time in seconds. Solve the equation for $\mathrm { t }$.

Accepted Answer

A)  $t = 2 \pi \cos V$ 
B)  $t = \frac { 1 } { 2 \pi } \cos \mathrm { V }$ 
C)  $t = \frac { 1 } { 2 \pi } \arccos V$ 
D)  $t = 2 \pi \arccos V$ 
A)  $t = 2 \pi \cos V$ 
B)  $t = \frac { 1 } { 2 \pi } \cos \mathrm { V }$ 
C)  $t = \frac { 1 } { 2 \pi } \arccos V$ 
D)  $t = 2 \pi \arccos V$

Question 8

Solve the equation for x, where x is restricted to the given interval.
-$y = 4 \cot \frac { x } { 2 }$, for $x$ in $( 0,2 \pi )$

Accepted Answer

A)  $x = 2 \operatorname { arccot } 4 y$ 
B)  $x = 2 \operatorname { arccot } \frac { y } { 4 }$ 
C)  $x = \frac { 1 } { 2 } \operatorname { arccot } 4 y$ 
D)  $x = \frac { 1 } { 2 } \operatorname { arccot } \frac { y } { 4 }$ 
A)  $x = 2 \operatorname { arccot } 4 y$ 
B)  $x = 2 \operatorname { arccot } \frac { y } { 4 }$ 
C)  $x = \frac { 1 } { 2 } \operatorname { arccot } 4 y$ 
D)  $x = \frac { 1 } { 2 } \operatorname { arccot } \frac { y } { 4 }$

Question 9

Rewrite the following as a product of trigonometric functions.
-$\sin 7 + \sin 43$

Accepted Answer

A)  $2 \sin 25 \cos ( - 18 )$ 
B)  $2 \cos 25 \sin ( - 18 )$ 
C)  $2 \sin 25 \sin ( - 18 )$ 
D)  $2 \cos 25 \cos ( - 18 )$ 
A)  $2 \sin 25 \cos ( - 18 )$ 
B)  $2 \cos 25 \sin ( - 18 )$ 
C)  $2 \sin 25 \sin ( - 18 )$ 
D)  $2 \cos 25 \cos ( - 18 )$

Question 10

Solve the equation for x, where x is restricted to the given interval.
-$y = 7 \sin x$, for $x$ in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$

Accepted Answer

A)  $x = \frac { 1 } { 7 } \arcsin y$ 
B)  $x = \arcsin 7 y$ 
C)  $x = 7 \arcsin y$ 
D)  $x = \arcsin \frac { y } { 7 }$ 
A)  $x = \frac { 1 } { 7 } \arcsin y$ 
B)  $x = \arcsin 7 y$ 
C)  $x = 7 \arcsin y$ 
D)  $x = \arcsin \frac { y } { 7 }$

Question 11

Use identities to write the expression as a single function of x or $$\theta$$
-$\sin \left( 360 ^ { \circ } + x \right)$

Accepted Answer

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

Question 12

Write the following as an algebraic expression in u, u > 0.
-$\cos ( \arctan \mathrm { u } )$

Accepted Answer

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

Question 13

Solve the problem.

-An airplane flying faster than the speed of sound creates sound waves that form a cone. If $\theta$ is the vertex angle of the cone and $\mathrm { m }$ is the Mach number for the speed of the plane, then $\sin \frac { \theta } { 2 } = \frac { 1 } { \mathrm {~m} }$ $( \mathrm { m } > 1 )$. Find the Mach number to the nearest tenth if $\theta = 60 ^ { \circ }$.

Accepted Answer

A)  2 
B)  $2.6$ 
C)  $3.1$ 
D)  $1.8$ 
A)  2 
B)  $2.6$ 
C)  $3.1$ 
D)  $1.8$

Question 14

Graph the inverse circular function.
-$y=\sin ^{-1} x$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 15

Use the fundamental identities to find the value of the trigonometric function.
-Find $\csc \theta$ if $\sin \theta = - \frac { 2 } { 3 }$ and $\theta$ is in quadrant IV.

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 16

Use a sum or difference identity to find the exact value.
-$$\sin \left( - \frac { 5 \pi } { 12 } \right)$$

Accepted Answer

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

Question 17

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

Accepted Answer

A)  1 
B)  $\tan ^ { 2 } \theta$ 
C)  $\csc ^ { 2 } \theta$ 
D)  $\sec ^ { 2 } \theta$ 
A)  1 
B)  $\tan ^ { 2 } \theta$ 
C)  $\csc ^ { 2 } \theta$ 
D)  $\sec ^ { 2 } \theta$

Question 18

Solve the equation for exact solutions.
-$\arctan x = \arcsin \frac { 7 } { 25 }$

Accepted Answer

A)  $\left\{ \frac { 7 } { 24 } \right\}$ 
B)  $\left\{ \frac { 24 } { 7 } \right\}$ 
C)  $\varnothing$ 
D)  $\left\{ \frac { 25 } { 24 } \right\}$ 
A)  $\left\{ \frac { 7 } { 24 } \right\}$ 
B)  $\left\{ \frac { 24 } { 7 } \right\}$ 
C)  $\varnothing$ 
D)  $\left\{ \frac { 25 } { 24 } \right\}$

Question 19

Use Identities to find the exact value.
-$$\cos 255 ^ { \circ }$$

Accepted Answer

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

Question 20

Find the exact value of the real number y.
-$$y = \sin ^ { - 1 } \left( \frac { \sqrt { 3 } } { 2 } \right)$$

Accepted Answer

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

Use identities to write the expression as a single function of x or $\theta$ - $\tan \left( 30 ^ { \circ } + x \right)$

Solve the equation for exact solutions. - $6 \arcsin x = \pi$

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

Find the exact value of the expression using the provided information. -Find $\tan ( \mathrm { s } + \mathrm { t } )$ given that $\sin \mathrm { s } = \frac { 1 } { 4 }$ , with $\mathrm { s }$ in quadrant II, and $\sin \mathrm { t } = - \frac { 1 } { 2 }$ , with $\mathrm { t }$ in quadrant IV.

Complete the sentence so the result is an identity. Let x be any real number. - $- 1 = \tan ^ { 2 } x$

Solve the equation for x, where x is restricted to the given interval. - $y = 4 \cot \frac { x } { 2 }$ , for $x$ in $( 0,2 \pi )$

Rewrite the following as a product of trigonometric functions. - $\sin 7 + \sin 43$

Solve the equation for x, where x is restricted to the given interval. - $y = 7 \sin x$ , for $x$ in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$

Use identities to write the expression as a single function of x or $\theta$ - $\sin \left( 360 ^ { \circ } + x \right)$

Write the following as an algebraic expression in u, u > 0. - $\cos ( \arctan \mathrm { u } )$

Graph the inverse circular function. - $y=\sin ^{-1} x$ $Graph the inverse circular function. - y=\sin ^{-1} x$

Use the fundamental identities to find the value of the trigonometric function. -Find $\csc \theta$ if $\sin \theta = - \frac { 2 } { 3 }$ and $\theta$ is in quadrant IV.

Use a sum or difference identity to find the exact value. - $\sin \left( - \frac { 5 \pi } { 12 } \right)$

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

Solve the equation for exact solutions. - $\arctan x = \arcsin \frac { 7 } { 25 }$

Use Identities to find the exact value. - $\cos 255 ^ { \circ }$

Find the exact value of the real number y. - $y = \sin ^ { - 1 } \left( \frac { \sqrt { 3 } } { 2 } \right)$

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

Use identities to write the expression as a single function of x or θ\thetaθ - tan⁡(30∘+x)\tan \left( 30 ^ { \circ } + x \right)tan(30∘+x)

Solve the equation for exact solutions. - 6arcsin⁡x=π6 \arcsin x = \pi6arcsinx=π

Give the degree measure of θ\thetaθ - θ=arcsin⁡(−12)\theta = \arcsin \left( - \frac { 1 } { 2 } \right)θ=arcsin(−21​)

Complete the sentence so the result is an identity. Let x be any real number. - −1=tan⁡2x- 1 = \tan ^ { 2 } x−1=tan2x

Solve the equation for x, where x is restricted to the given interval. - y=4cot⁡x2y = 4 \cot \frac { x } { 2 }y=4cot2x​ , for xxx in (0,2π)( 0,2 \pi )(0,2π)

Rewrite the following as a product of trigonometric functions. - sin⁡7+sin⁡43\sin 7 + \sin 43sin7+sin43

Solve the equation for x, where x is restricted to the given interval. - y=7sin⁡xy = 7 \sin xy=7sinx , for xxx in [−π2,π2]\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right][−2π​,2π​]

Use identities to write the expression as a single function of x or θ\thetaθ - sin⁡(360∘+x)\sin \left( 360 ^ { \circ } + x \right)sin(360∘+x)

Write the following as an algebraic expression in u, u > 0. - cos⁡(arctan⁡u)\cos ( \arctan \mathrm { u } )cos(arctanu)

Graph the inverse circular function. - y=sin⁡−1xy=\sin ^{-1} xy=sin−1x

Use the fundamental identities to find the value of the trigonometric function. -Find csc⁡θ\csc \thetacscθ if sin⁡θ=−23\sin \theta = - \frac { 2 } { 3 }sinθ=−32​ and θ\thetaθ is in quadrant IV.

Use a sum or difference identity to find the exact value. - sin⁡(−5π12)\sin \left( - \frac { 5 \pi } { 12 } \right)sin(−125π​)

Use the fundamental identities to simplify the expression. - 1cot⁡2θ+sec⁡θcos⁡θ\frac { 1 } { \cot ^ { 2 } \theta } + \sec \theta \cos \thetacot2θ1​+secθcosθ

Solve the equation for exact solutions. - arctan⁡x=arcsin⁡725\arctan x = \arcsin \frac { 7 } { 25 }arctanx=arcsin257​

Use Identities to find the exact value. - cos⁡255∘\cos 255 ^ { \circ }cos255∘

Find the exact value of the real number y. - y=sin⁡−1(32)y = \sin ^ { - 1 } \left( \frac { \sqrt { 3 } } { 2 } \right)y=sin−1(23​​)