Question 1

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

Accepted Answer

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

Question 2

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 $m$ is the Mach number for the speed of the plane, then $\sin \frac { \theta } { 2 } = \frac { 1 } { m }$ $( \mathrm { m } > 1 )$. Find $\theta$ to the nearest degree if $\mathrm { m } = 3.9$.

Accepted Answer

A)  $35 ^ { \circ }$ 
B)  $33 ^ { \circ }$ 
C)  $30 ^ { \circ }$ 
D)  $28 ^ { \circ }$ 
A)  $35 ^ { \circ }$ 
B)  $33 ^ { \circ }$ 
C)  $30 ^ { \circ }$ 
D)  $28 ^ { \circ }$

Question 3

Use identities to find the indicated value for each angle measure.
-$\tan \theta = \frac { 7 } { 24 } , \quad \sin \theta < 0 \quad$ Find $\cos ( 2 \theta )$

Accepted Answer

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

Question 4

Solve the equation for x, where x is restricted to the given interval.
-$y = 3 \tan 2 x - 1$, for $x$ in $\left( - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$

Accepted Answer

A)  $x = \frac { 1 } { 2 } \arctan \frac { y + 1 } { 3 }$ 
B)  $x = 2 \arctan \frac { y + 1 } { 3 }$ 
C)  $x = \frac { 1 } { 2 } \arctan ( 3 y + 3 )$ 
D)  $x = 2 \arctan ( 3 y + 3 )$ 
A)  $x = \frac { 1 } { 2 } \arctan \frac { y + 1 } { 3 }$ 
B)  $x = 2 \arctan \frac { y + 1 } { 3 }$ 
C)  $x = \frac { 1 } { 2 } \arctan ( 3 y + 3 )$ 
D)  $x = 2 \arctan ( 3 y + 3 )$

Question 5

Solve the equation for exact solutions.
-The formula for the up-and-down motion of a weight on a spring is given by
$$\mathrm { S } = \mathrm { a } \sin \left( \frac { \sqrt { \mathrm { k } } } { \mathrm { m } } \right) \mathrm { t }$$
where $a$ is the radius of the circle, $\mathrm { k }$ is the spring constant, $\mathrm { m }$ is the mass, and $\mathrm { t }$ is the time. Solve the equation for $\mathrm { t }$.

Accepted Answer

A)  $t = \frac { m } { \sqrt { k } } \sin \left( \frac { S } { a } \right)$ 
B)  $t = \frac { \sqrt { \mathrm { k } } } { \mathrm { m } } \arcsin \left( \frac { \mathrm { s } } { \mathrm { a } } \right)$ 
C)  $t = \frac { m } { \sqrt { \mathrm { k } } } \arcsin \left( \frac { \mathrm { s } } { \mathrm { a } } \right)$ 
D)  $t = \frac { \sqrt { \mathrm { k } } } { \mathrm { m } } \sin \left( \frac { \mathrm { a } } { \mathrm { S } } \right)$ 
A)  $t = \frac { m } { \sqrt { k } } \sin \left( \frac { S } { a } \right)$ 
B)  $t = \frac { \sqrt { \mathrm { k } } } { \mathrm { m } } \arcsin \left( \frac { \mathrm { s } } { \mathrm { a } } \right)$ 
C)  $t = \frac { m } { \sqrt { \mathrm { k } } } \arcsin \left( \frac { \mathrm { s } } { \mathrm { a } } \right)$ 
D)  $t = \frac { \sqrt { \mathrm { k } } } { \mathrm { m } } \sin \left( \frac { \mathrm { a } } { \mathrm { S } } \right)$

Question 6

Provide an appropriate response.
-The equation $\sin 3 x - \cos 3 x = 3$ has no solution in the interval $[ 0,2 \pi )$. Explain what this tells you about the graph of $y = \sin 3 x - \cos 3 x - 3$.

Accepted Answer

The graph

Question 7

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

Question 8

Find the exact value of the expression using the provided information.
-Find $\sin ( \mathrm { s } + \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 5 } { 13 }$, with $\mathrm { s }$ in quadrant II, and $\sin \mathrm { t } = \frac { 15 } { 17 }$, with $\mathrm { t }$ in quadrant II.

Accepted Answer

A)  $\frac { 21 } { 221 }$ 
B)  $\frac { 171 } { 221 }$ 
C)  $- \frac { 21 } { 221 }$ 
D)  $- \frac { 171 } { 221 }$ 
A)  $\frac { 21 } { 221 }$ 
B)  $\frac { 171 } { 221 }$ 
C)  $- \frac { 21 } { 221 }$ 
D)  $- \frac { 171 } { 221 }$

Question 9

Decide whether the expression is or is not an identity.
-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)  $\mu _ { \mathrm { S } } = \cot \theta _ { \mathrm { S } }$ 
B)  $\mu _ { \mathrm { S } } = \sec \theta _ { \mathrm { S } }$ 
C)  $\mu _ { \mathrm { s } } = \tan ^ { 2 } \theta _ { \mathrm { s } }$ 
D)  $\mu _ { \mathrm { S } } = \tan \theta _ { \mathrm { S } }$ 
A)  $\mu _ { \mathrm { S } } = \cot \theta _ { \mathrm { S } }$ 
B)  $\mu _ { \mathrm { S } } = \sec \theta _ { \mathrm { S } }$ 
C)  $\mu _ { \mathrm { s } } = \tan ^ { 2 } \theta _ { \mathrm { s } }$ 
D)  $\mu _ { \mathrm { S } } = \tan \theta _ { \mathrm { S } }$

Question 10

Use the fundamental identities to find the value of the trigonometric function.
-Find $\cos \theta$ if $\csc \theta = - \frac { 5 } { 3 }$ and $\theta$ is in quadrant IV.

Accepted Answer

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

Question 11

Find the exact value of the expression using the provided information.
-Find $\cos ( s + t )$ given that $\cos s = - \frac { 1 } { 2 }$, with $s$ in quadrant III, and $\cos t = - \frac { 3 } { 5 }$, with $t$ in quadrant III.

Accepted Answer

A)  $\frac { - 3 - 4 \sqrt { 3 } } { 10 }$ 
B)  $\frac { 3 - 4 \sqrt { 3 } } { 10 }$ 
C)  $\frac { 3 - 4 \sqrt { 5 } } { 10 }$ 
D)  $\frac { 3 + 4 \sqrt { 3 } } { 10 }$ 
A)  $\frac { - 3 - 4 \sqrt { 3 } } { 10 }$ 
B)  $\frac { 3 - 4 \sqrt { 3 } } { 10 }$ 
C)  $\frac { 3 - 4 \sqrt { 5 } } { 10 }$ 
D)  $\frac { 3 + 4 \sqrt { 3 } } { 10 }$

Question 12

Use the fundamental identities to find the value of the trigonometric function.
-Find $\sin \theta$ if $\cos \theta = - \frac { \sqrt { 11 } } { 6 }$ and $\theta$ is in quadrant III.

Accepted Answer

A)  $- \frac { \sqrt { 3 } } { 12 }$ 
B)  $\frac { 5 } { 6 }$ 
C)  $- \frac { 25 } { 36 }$ 
D)  $- \frac { 5 } { 6 }$ 
A)  $- \frac { \sqrt { 3 } } { 12 }$ 
B)  $\frac { 5 } { 6 }$ 
C)  $- \frac { 25 } { 36 }$ 
D)  $- \frac { 5 } { 6 }$

Question 13

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

Accepted Answer

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

Question 14

Solve the equation for exact solutions.
-$\arcsin \left( y - \frac { \pi } { 6 } \right) = \frac { \pi } { 6 }$

Accepted Answer

A)  $\left\{ \frac { 3 - \pi } { 6 } \right\}$ 
B)  $\varnothing$ 
C)  $\left\{ \frac { 3 + \pi } { 3 } \right\}$ 
D)  $\left\{ \frac { 3 + \pi } { 6 } \right\}$ 
A)  $\left\{ \frac { 3 - \pi } { 6 } \right\}$ 
B)  $\varnothing$ 
C)  $\left\{ \frac { 3 + \pi } { 3 } \right\}$ 
D)  $\left\{ \frac { 3 + \pi } { 6 } \right\}$

Question 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

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.
-$$\cos ^ { 2 } x - 1 = 0$$

Accepted Answer

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

Question 18

Write the following as an algebraic expression in u, u > 0.
-$$\sin \left( \operatorname { arcsec } \frac { \sqrt { u ^ { 2 } + 4 } } { u } \right)$$

Accepted Answer

A)  $u \sqrt { 2 }$ 
B)  $\frac { 2 \sqrt { \mathrm { u } ^ { 2 } + 4 } } { \mathrm { u } ^ { 2 } + 4 }$ 
C)  $\frac { \mathrm { u } \sqrt { \mathrm { u } ^ { 2 } + 2 } } { \mathrm { u } ^ { 2 } + 2 }$ 
D)  $\frac { \sqrt { u ^ { 2 } + 2 } } { u ^ { 2 } + 2 }$ 
A)  $u \sqrt { 2 }$ 
B)  $\frac { 2 \sqrt { \mathrm { u } ^ { 2 } + 4 } } { \mathrm { u } ^ { 2 } + 4 }$ 
C)  $\frac { \mathrm { u } \sqrt { \mathrm { u } ^ { 2 } + 2 } } { \mathrm { u } ^ { 2 } + 2 }$ 
D)  $\frac { \sqrt { u ^ { 2 } + 2 } } { u ^ { 2 } + 2 }$

Question 19

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-$\tan x ( \cot x - \cos x )$

Accepted Answer

A)  $- \sec ^ { 2 } x$ 
B)  $1 - \sin x$ 
C)  0 
D)  1 
A)  $- \sec ^ { 2 } x$ 
B)  $1 - \sin x$ 
C)  0 
D)  1

Question 20

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

Accepted Answer

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

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

Use identities to find the indicated value for each angle measure. - $\tan \theta = \frac { 7 } { 24 } , \quad \sin \theta < 0 \quad$ Find $\cos ( 2 \theta )$

Solve the equation for x, where x is restricted to the given interval. - $y = 3 \tan 2 x - 1$ , for $x$ in $\left( - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$

Provide an appropriate response. -The equation $\sin 3 x - \cos 3 x = 3$ has no solution in the interval $[ 0,2 \pi )$ . Explain what this tells you about the graph of $y = \sin 3 x - \cos 3 x - 3$ .

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

Find the exact value of the expression using the provided information. -Find $\sin ( \mathrm { s } + \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 5 } { 13 }$ , with $\mathrm { s }$ in quadrant II, and $\sin \mathrm { t } = \frac { 15 } { 17 }$ , with $\mathrm { t }$ in quadrant II.

Use the fundamental identities to find the value of the trigonometric function. -Find $\cos \theta$ if $\csc \theta = - \frac { 5 } { 3 }$ and $\theta$ is in quadrant IV.

Find the exact value of the expression using the provided information. -Find $\cos ( s + t )$ given that $\cos s = - \frac { 1 } { 2 }$ , with $s$ in quadrant III, and $\cos t = - \frac { 3 } { 5 }$ , with $t$ in quadrant III.

Use the fundamental identities to find the value of the trigonometric function. -Find $\sin \theta$ if $\cos \theta = - \frac { \sqrt { 11 } } { 6 }$ and $\theta$ is in quadrant III.

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

Solve the equation for exact solutions. - $\arcsin \left( y - \frac { \pi } { 6 } \right) = \frac { \pi } { 6 }$

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

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

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. - $\cos ^ { 2 } x - 1 = 0$

Write the following as an algebraic expression in u, u > 0. - $\sin \left( \operatorname { arcsec } \frac { \sqrt { u ^ { 2 } + 4 } } { u } \right)$

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\tan x ( \cot x - \cos x )$

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

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

Complete the sentence so the result is an identity. Let x be any real number. - −=1sin⁡x- = \frac { 1 } { \sin x }−=sinx1​

Use identities to find the indicated value for each angle measure. - 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)

Solve the equation for x, where x is restricted to the given interval. - y=3tan⁡2x−1y = 3 \tan 2 x - 1y=3tan2x−1 , for xxx in (−π4,π4)\left( - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)(−4π​,4π​)

Provide an appropriate response. -The equation sin⁡3x−cos⁡3x=3\sin 3 x - \cos 3 x = 3sin3x−cos3x=3 has no solution in the interval [0,2π)[ 0,2 \pi )[0,2π) . Explain what this tells you about the graph of y=sin⁡3x−cos⁡3x−3y = \sin 3 x - \cos 3 x - 3y=sin3x−cos3x−3 .

Use a sum or difference identity to find the exact value. - sin⁡185∘cos⁡65∘−cos⁡185∘sin⁡65∘\sin 185 ^ { \circ } \cos 65 ^ { \circ } - \cos 185 ^ { \circ } \sin 65 ^ { \circ }sin185∘cos65∘−cos185∘sin65∘

Use the fundamental identities to find the value of the trigonometric function. -Find cos⁡θ\cos \thetacosθ if csc⁡θ=−53\csc \theta = - \frac { 5 } { 3 }cscθ=−35​ and θ\thetaθ is in quadrant IV.

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=−12\cos s = - \frac { 1 } { 2 }coss=−21​ , with sss in quadrant III, and cos⁡t=−35\cos t = - \frac { 3 } { 5 }cost=−53​ , with ttt in quadrant III.

Use the fundamental identities to find the value of the trigonometric function. -Find sin⁡θ\sin \thetasinθ if cos⁡θ=−116\cos \theta = - \frac { \sqrt { 11 } } { 6 }cosθ=−611​​ and θ\thetaθ is in quadrant III.

Solve the equation for exact solutions over the interval [0, 2 [0,2π)[ 0,2 \pi )[0,2π) - 2sin⁡2x=sin⁡x2 \sin ^ { 2 } x = \sin x2sin2x=sinx

Solve the equation for exact solutions. - arcsin⁡(y−π6)=π6\arcsin \left( y - \frac { \pi } { 6 } \right) = \frac { \pi } { 6 }arcsin(y−6π​)=6π​

Find the exact value by using a half-angle identity. - cos⁡75∘\cos 75 ^ { \circ }cos75∘

Find the exact value by using a half-angle identity. - cos⁡165∘\cos 165 ^ { \circ }cos165∘

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. - cos⁡2x−1=0\cos ^ { 2 } x - 1 = 0cos2x−1=0

Write the following as an algebraic expression in u, u > 0. - sin⁡(arcsec⁡u2+4u)\sin \left( \operatorname { arcsec } \frac { \sqrt { u ^ { 2 } + 4 } } { u } \right)sin(arcsecuu2+4​​)

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - tan⁡x(cot⁡x−cos⁡x)\tan x ( \cot x - \cos x )tanx(cotx−cosx)

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