Question 1

Write in terms of the cofunction of a complementary angle.
-$\sec 19 ^ { \circ } 8 ^ { \prime }$

Accepted Answer

A)  $\cos 160 ^ { \circ } 52 ^ { \prime }$ 
B)  $\cos 70 ^ { \circ } 52 ^ { \prime }$ 
C)  $\csc 71 ^ { \circ } 8 ^ { \prime }$ 
D)  $\csc 70 ^ { \circ } 52 ^ { \prime }$ 
A)  $\cos 160 ^ { \circ } 52 ^ { \prime }$ 
B)  $\cos 70 ^ { \circ } 52 ^ { \prime }$ 
C)  $\csc 71 ^ { \circ } 8 ^ { \prime }$ 
D)  $\csc 70 ^ { \circ } 52 ^ { \prime }$

Question 2

Decide whether the expression is or is not an identity.
-The power dissipated in an electric circuit is given by the expression $\mathrm { P } = \mathrm { RI } ^ { 2 }$, where $\mathrm { R }$ is the resistance of the circuit and $\mathrm { I }$ is the current through the circuit. For a sinusoidal alternating current, the current might be represented by the relation $\mathrm { I } = \mathrm { A } \sin ( 2 \pi \mathrm { ft } )$, where $\mathrm { A }$ is the amplitude, $\mathrm { f }$ is the frequency, and $\mathrm { t }$ is time. Write an expression for $\mathrm { P }$ involving the sine function, and use a fundamental identity to write $\mathrm { P }$ in terms of the cosine function.

Accepted Answer

A)  $P = R A \sin ^ { 2 } ( 2 \pi f t ) ; P = R A - R A \cos ^ { 2 } ( 2 \pi f t )$ 
B)  $\mathrm { P } = \mathrm { RA } ^ { 2 } \sin ^ { 2 } ( 2 \pi \mathrm { ft } ) ; \mathrm { P } = - \mathrm { RA } ^ { 2 } \cos ^ { 2 } ( 2 \pi \mathrm { ft } )$ 
C)  $P = R A \sin ^ { 2 } ( 2 \pi \mathrm { ft } ) ; P = R A - \cos ^ { 2 } ( 2 \pi f t )$ 
D)  $\mathrm { P } = \mathrm { RA } ^ { 2 } \sin ^ { 2 } ( 2 \pi \mathrm { ft } ) ; \mathrm { P } = \mathrm { RA } ^ { 2 } - \mathrm { RA } ^ { 2 } \cos ^ { 2 } ( 2 \pi \mathrm { ft } )$ 
A)  $P = R A \sin ^ { 2 } ( 2 \pi f t ) ; P = R A - R A \cos ^ { 2 } ( 2 \pi f t )$ 
B)  $\mathrm { P } = \mathrm { RA } ^ { 2 } \sin ^ { 2 } ( 2 \pi \mathrm { ft } ) ; \mathrm { P } = - \mathrm { RA } ^ { 2 } \cos ^ { 2 } ( 2 \pi \mathrm { ft } )$ 
C)  $P = R A \sin ^ { 2 } ( 2 \pi \mathrm { ft } ) ; P = R A - \cos ^ { 2 } ( 2 \pi f t )$ 
D)  $\mathrm { P } = \mathrm { RA } ^ { 2 } \sin ^ { 2 } ( 2 \pi \mathrm { ft } ) ; \mathrm { P } = \mathrm { RA } ^ { 2 } - \mathrm { RA } ^ { 2 } \cos ^ { 2 } ( 2 \pi \mathrm { ft } )$

Question 3

Use the fundamental identities to find the value of the trigonometric function.
-Find $\cos \theta$ if $\tan \theta = \frac { 2 } { 5 }$ and $\theta$ is in quadrant III.

Accepted Answer

A)  $\frac { \sqrt { 33 } } { 4 }$ 
B)  $- \frac { \sqrt { 11 } } { 6 }$ 
C)  $\frac { 4 } { 5 }$ 
D)  $- \frac { 5 \sqrt { 29 } } { 29 }$ 
A)  $\frac { \sqrt { 33 } } { 4 }$ 
B)  $- \frac { \sqrt { 11 } } { 6 }$ 
C)  $\frac { 4 } { 5 }$ 
D)  $- \frac { 5 \sqrt { 29 } } { 29 }$

Question 4

Use identities to write the expression as a single function of x or $$\theta$$
-$\cos ( \theta - \pi )$

Accepted Answer

A)  $- \sin \theta$ 
B)  $\sin \theta$ 
C)  $\cos \theta$ 
D)  $- \cos \theta$ 
A)  $- \sin \theta$ 
B)  $\sin \theta$ 
C)  $\cos \theta$ 
D)  $- \cos \theta$

Question 5

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.
-

Accepted Answer

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

Question 6

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi )$$
-$\csc 3 x = 0$

Accepted Answer

A)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
B)  $\varnothing$ 
C)  $\left\{ \frac { \pi } { 8 } , \frac { 9 \pi } { 8 } \right\}$ 
D)  $\left\{ 0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } \right\}$ 
A)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
B)  $\varnothing$ 
C)  $\left\{ \frac { \pi } { 8 } , \frac { 9 \pi } { 8 } \right\}$ 
D)  $\left\{ 0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } \right\}$

Question 7

Verify that each equation is an identity.
-$$\tan x ( \csc x - \sin x ) = \cos x$$

Accepted Answer

\[\begin{array} { l } 
\tan x ( \csc x -

Question 8

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi )$$
-$$\sec \frac { x } { 2 } = \cos \frac { x } { 2 }$$

Accepted Answer

A)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
B)  $\{ 0 \}$ 
C)  $\left\{ 0 , \frac { \pi } { 4 } , \pi , \frac { 5 \pi } { 3 } \right\}$ 
D)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } \right\}$ 
A)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
B)  $\{ 0 \}$ 
C)  $\left\{ 0 , \frac { \pi } { 4 } , \pi , \frac { 5 \pi } { 3 } \right\}$ 
D)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } \right\}$

Question 9

Use the fundamental identities to find the value of the trigonometric function.
-Find $\sin \theta$ if $\cot \theta = - 6$ and $\cos \theta < 0$.

Accepted Answer

A)  $- 37$ 
B)  $\frac { \sqrt { 37 } } { 6 }$ 
C)  $- \frac { 1 } { 6 }$ 
D)  $\frac { \sqrt { 37 } } { 37 }$ 
A)  $- 37$ 
B)  $\frac { \sqrt { 37 } } { 6 }$ 
C)  $- \frac { 1 } { 6 }$ 
D)  $\frac { \sqrt { 37 } } { 37 }$

Question 10

Use the fundamental identities to find the value of the trigonometric function.
-Find $\tan \theta$ if $\sec \theta = \frac { \sqrt { 10 } } { 3 }$ and $\sin \theta < 0$

Accepted Answer

A)  $- 3$ 
B)  3 
C)  $- \frac { 1 } { 3 }$ 
D)  $- \sqrt { 10 }$ 
A)  $- 3$ 
B)  3 
C)  $- \frac { 1 } { 3 }$ 
D)  $- \sqrt { 10 }$

Question 11

Solve the equation for solutions in the interval [0 $$\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$$ ). Round to the nearest degree.
-$\sin 2 \theta + \sin \theta = 0$

Accepted Answer

A)  $\left\{ 0 ^ { \circ } , 120 ^ { \circ } , 180 ^ { \circ } , 240 ^ { \circ } \right\}$ 
B)  $\left\{ 105 ^ { \circ } , 165 ^ { \circ } , 285 ^ { \circ } , 345 ^ { \circ } \right\}$ 
C)  $\left\{ 30 ^ { \circ } , 90 ^ { \circ } , 150 ^ { \circ } , 270 ^ { \circ } \right\}$ 
D)  $\left\{ 15 ^ { \circ } , 165 ^ { \circ } , 195 ^ { \circ } , 345 ^ { \circ } \right\}$ 
A)  $\left\{ 0 ^ { \circ } , 120 ^ { \circ } , 180 ^ { \circ } , 240 ^ { \circ } \right\}$ 
B)  $\left\{ 105 ^ { \circ } , 165 ^ { \circ } , 285 ^ { \circ } , 345 ^ { \circ } \right\}$ 
C)  $\left\{ 30 ^ { \circ } , 90 ^ { \circ } , 150 ^ { \circ } , 270 ^ { \circ } \right\}$ 
D)  $\left\{ 15 ^ { \circ } , 165 ^ { \circ } , 195 ^ { \circ } , 345 ^ { \circ } \right\}$

Question 12

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as
appropriate.
-It can be shown that if the angle of elevation from an observer to the top of an object is $\mathrm { A }$ and the an: elevation $\mathrm { d } \mathrm { ft }$ closer is $\mathrm { B }$, then the distance from the object to the closest point of observation is given $\mathrm { D } = \frac { \mathrm { d } \cot \mathrm { B } } { \cot \mathrm { A } - \cot \mathrm { B } } \mathrm { ft }$.
Find $\mathrm { B }$ if $\mathrm { D } = 70 \mathrm { ft } , \mathrm { d } = 30 \mathrm { ft }$, and $\mathrm { A } = 39 ^ { \circ }$. Give your answer in degrees to the nearest hundredth.

Accepted Answer

A)  $39.00 ^ { \circ }$ 
B)  $49.16 ^ { \circ }$ 
C)  $19.14 ^ { \circ }$ 
D)  $69.67 ^ { \circ }$ ngle of451) 
A)  $39.00 ^ { \circ }$ 
B)  $49.16 ^ { \circ }$ 
C)  $19.14 ^ { \circ }$ 
D)  $69.67 ^ { \circ }$ ngle of451)

Question 13

Find the exact value of the expression using the provided information.
-Find $\tan ( \mathrm { s } - \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 5 } { 13 }$, with $\mathrm { s }$ in quadrant $\mathrm { II }$, and $\sin \mathrm { t } = \frac { 8 } { 17 }$, with $\mathrm { t }$ in quadrant II.

Accepted Answer

A)  $\frac { 20 } { 3 }$ 
B)  $- \frac { 140 } { 171 }$ 
C)  $- \frac { 220 } { 171 }$ 
D)  $- \frac { 20 } { 3 }$ 
A)  $\frac { 20 } { 3 }$ 
B)  $- \frac { 140 } { 171 }$ 
C)  $- \frac { 220 } { 171 }$ 
D)  $- \frac { 20 } { 3 }$

Question 14

Determine all solutions of the equation in radians.
-Find $\sin \frac { \theta } { 2 }$, given that $\cos \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in $0 < \theta < 90 ^ { \circ }$.

Accepted Answer

A)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
B)  $\frac { \sqrt { 10 } } { 4 }$ 
C)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
D)  $\frac { \sqrt { 6 } } { 4 }$ 
A)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
B)  $\frac { \sqrt { 10 } } { 4 }$ 
C)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
D)  $\frac { \sqrt { 6 } } { 4 }$

Question 15

Verify that the equation is an identity.
-Two sound waves, producing the pressures $P _ { 1 } = 4 \sin ( 400 \pi t )$ and $P _ { 2 } = 3 \cos ( 400 \pi t )$, combine on a person's ear drum to produce the pressure $\mathrm { P } = \mathrm { P } _ { 1 } + \mathrm { P } _ { 2 }$. Graph $\mathrm { P }$ in $[ - 0.01,0.01,0.001 ]$ by $[ - 20,20,2 ]$ and use the graph to estimate values for $\mathrm { a }$ and $\mathrm { k }$ so that $\mathrm { P } = \mathrm { a } \sin ( 400 \pi \mathrm { t } + \mathrm { k } )$. Round $\mathrm { k }$ to the nearest ten-thousandth, if necessary.

Accepted Answer

A)  $\mathrm { a } = 2.2 ; \mathrm { k } = 0.9273$ 
B)  $\mathrm { a } = 5 ; \mathrm { k } = 0.6435$ 
C)  $a = 5 ; k = 808.6467$ 
D)  $\mathrm { a } = 2.2 ; \mathrm { k } = 808.6467$ 
A)  $\mathrm { a } = 2.2 ; \mathrm { k } = 0.9273$ 
B)  $\mathrm { a } = 5 ; \mathrm { k } = 0.6435$ 
C)  $a = 5 ; k = 808.6467$ 
D)  $\mathrm { a } = 2.2 ; \mathrm { k } = 808.6467$

Question 16

Write in terms of the cofunction of a complementary angle.
-$\tan 173 ^ { \circ } 12 ^ { \prime }$

Accepted Answer

A)  $\cot 83 ^ { \circ } 48 ^ { \prime }$ 
B)  $\tan - 83 ^ { \circ } 48 ^ { \prime }$ 
C)  $\cot - 83 ^ { \circ } 12 ^ { \prime }$ 
D)  $\cot 82 ^ { \circ } 48 ^ { \prime }$ 
A)  $\cot 83 ^ { \circ } 48 ^ { \prime }$ 
B)  $\tan - 83 ^ { \circ } 48 ^ { \prime }$ 
C)  $\cot - 83 ^ { \circ } 12 ^ { \prime }$ 
D)  $\cot 82 ^ { \circ } 48 ^ { \prime }$

Question 17

Find the exact value by using a half-angle identity.
-$\tan 75 ^ { \circ }$

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 18

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

Accepted Answer

The answer of Verify that the equation is an identity.
-\[\tan...

Question 19

Write the product as a sum or difference of trigonometric functions.
-$2 \cos 8 x \cos 2 x$

Accepted Answer

A)  $\frac { 1 } { 2 } ( \cos 10 x + \cos 6 x )$ 
B)  $\cos 6 x - \cos 10 x$ 
C)  $\sin 10 x + \sin 6 x$ 
D)  $\cos 10 x + \cos 6 x$ 
A)  $\frac { 1 } { 2 } ( \cos 10 x + \cos 6 x )$ 
B)  $\cos 6 x - \cos 10 x$ 
C)  $\sin 10 x + \sin 6 x$ 
D)  $\cos 10 x + \cos 6 x$

Question 20

Use the fundamental identities to simplify the expression.
-$$\frac { 1 + \tan ^ { 2 } x } { \sec x }$$

Accepted Answer

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

Write in terms of the cofunction of a complementary angle. - $\sec 19 ^ { \circ } 8 ^ { \prime }$

Use the fundamental identities to find the value of the trigonometric function. -Find $\cos \theta$ if $\tan \theta = \frac { 2 } { 5 }$ and $\theta$ is in quadrant III.

Use identities to write the expression as a single function of x or $\theta$ - $\cos ( \theta - \pi )$

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. -

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi )$ - $\csc 3 x = 0$

Verify that each equation is an identity. - $\tan x ( \csc x - \sin x ) = \cos x$

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi )$ - $\sec \frac { x } { 2 } = \cos \frac { x } { 2 }$

Use the fundamental identities to find the value of the trigonometric function. -Find $\sin \theta$ if $\cot \theta = - 6$ and $\cos \theta < 0$ .

Use the fundamental identities to find the value of the trigonometric function. -Find $\tan \theta$ if $\sec \theta = \frac { \sqrt { 10 } } { 3 }$ and $\sin \theta < 0$

Solve the equation for solutions in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ ). Round to the nearest degree. - $\sin 2 \theta + \sin \theta = 0$

Determine all solutions of the equation in radians. -Find $\sin \frac { \theta } { 2 }$ , given that $\cos \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in $0 < \theta < 90 ^ { \circ }$ .

Write in terms of the cofunction of a complementary angle. - $\tan 173 ^ { \circ } 12 ^ { \prime }$

Find the exact value by using a half-angle identity. - $\tan 75 ^ { \circ }$

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

Write the product as a sum or difference of trigonometric functions. - $2 \cos 8 x \cos 2 x$

Use the fundamental identities to simplify the expression. - $\frac { 1 + \tan ^ { 2 } x } { \sec x }$

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

Write in terms of the cofunction of a complementary angle. - sec⁡19∘8′\sec 19 ^ { \circ } 8 ^ { \prime }sec19∘8′

Use the fundamental identities to find the value of the trigonometric function. -Find cos⁡θ\cos \thetacosθ if tan⁡θ=25\tan \theta = \frac { 2 } { 5 }tanθ=52​ and θ\thetaθ is in quadrant III.

Use identities to write the expression as a single function of x or θ\thetaθ - cos⁡(θ−π)\cos ( \theta - \pi )cos(θ−π)