Question 1

Solve the equation for solutions in the interval [0, 2 π).
-$$\sin 4 x = \frac { \sqrt { 3 } } { 2 }$$

Accepted Answer

A)  $\left\{ 0 , \frac { \pi } { 4 } , \pi \right\}$ 
B)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } , \frac { 19 \pi } { 12 } \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
D)  $\{ 0 \}$ 
A)  $\left\{ 0 , \frac { \pi } { 4 } , \pi \right\}$ 
B)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } , \frac { 19 \pi } { 12 } \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
D)  $\{ 0 \}$

Question 2

Solve the equation for x, where x is restricted to the given interval.
-$$y = \cos ( x + 12 ) \text {, for } x \text { in } [ - 12 , \pi - 12 ]$$

Accepted Answer

A)  $x = \arccos y + 12$ 
B)  $x = \arccos ( y - 12 )$ 
C)  $x = \arccos y - 12$ 
D)  $x = \arccos ( y + 12 )$ 
A)  $x = \arccos y + 12$ 
B)  $x = \arccos ( y - 12 )$ 
C)  $x = \arccos y - 12$ 
D)  $x = \arccos ( y + 12 )$

Question 3

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary.
-sec (arctan 4.839)

Accepted Answer

A)  $0.2024$ 
B)  $4.9412$ 
C)  1 
D)  $0.5$ 
A)  $0.2024$ 
B)  $4.9412$ 
C)  1 
D)  $0.5$

Question 4

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

Accepted Answer

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

Question 5

Use the fundamental identities to find the value of the trigonometric function.
-Find $\sin \theta$ if $\cos \theta = \frac { 5 } { 6 }$ and $\theta$ is in quadrant IV.

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

Use a sum or difference identity to find the exact value.
-$\sin \frac { 7 \pi } { 12 }$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$ 
C)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
D)  $\frac { \sqrt { 3 } + 1 } { 2 }$ 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$ 
C)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
D)  $\frac { \sqrt { 3 } + 1 } { 2 }$

Question 8

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree.
-$\sqrt { 3 } \sec 2 \theta = 2$

Accepted Answer

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

Question 9

Write the product as a sum or difference of trigonometric functions.
-$4 \cos 16 ^ { \circ } \cos 8 ^ { \circ }$

Accepted Answer

A)  $2 \left( \sin 24 ^ { \circ } - \sin 8 ^ { \circ } \right)$ 
B)  $4 \left( \sin 24 ^ { \circ } + \sin 8 ^ { \circ } \right)$ 
C)  $2 \left( \cos 24 ^ { \circ } + \cos 8 ^ { \circ } \right)$ 
D)  $2 \left( \cos 24 ^ { \circ } - \cos 8 ^ { \circ } \right)$ 
A)  $2 \left( \sin 24 ^ { \circ } - \sin 8 ^ { \circ } \right)$ 
B)  $4 \left( \sin 24 ^ { \circ } + \sin 8 ^ { \circ } \right)$ 
C)  $2 \left( \cos 24 ^ { \circ } + \cos 8 ^ { \circ } \right)$ 
D)  $2 \left( \cos 24 ^ { \circ } - \cos 8 ^ { \circ } \right)$

Question 10

Give the exact value of the expression.
-$\sin ( \arctan 2 )$

Accepted Answer

A)  $\frac { 5 \sqrt { 2 } } { 2 }$ 
B)  $5 \sqrt { 2 }$ 
C)  $\frac { 2 \sqrt { 5 } } { 5 }$ 
D)  $2 \sqrt { 5 }$ 
A)  $\frac { 5 \sqrt { 2 } } { 2 }$ 
B)  $5 \sqrt { 2 }$ 
C)  $\frac { 2 \sqrt { 5 } } { 5 }$ 
D)  $2 \sqrt { 5 }$

Question 11

Solve the equation for x, where x is restricted to the given interval.
-$y = 8 \cos 3 x$, for $x$ in $\left[ 0 , \frac { \pi } { 3 } \right]$

Accepted Answer

A)  $x = 8 \arccos \frac { y } { 3 }$ 
B)  $x = \frac { 1 } { 8 } \arccos \frac { y } { 3 }$ 
C)  $x = 3 \arccos \frac { y } { 8 }$ 
D)  $x = \frac { 1 } { 3 } \arccos \frac { y } { 8 }$ 
A)  $x = 8 \arccos \frac { y } { 3 }$ 
B)  $x = \frac { 1 } { 8 } \arccos \frac { y } { 3 }$ 
C)  $x = 3 \arccos \frac { y } { 8 }$ 
D)  $x = \frac { 1 } { 3 } \arccos \frac { y } { 8 }$

Question 12

Verify that the equation is an identity.
-$$\sin \left( x + \frac { \pi } { 2 } \right) = \cos x$$

Accepted Answer

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

Question 13

Find the exact value of the expression using the provided information.
-Find $\tan ( \mathrm { s } + \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 12 } { 13 }$, with $\mathrm { s }$ in quadrant II, and $\sin t = \frac { 8 } { 17 }$, with $t$ in quadrant II.

Accepted Answer

A)  $- \frac { 171 } { 221 }$ 
B)  $- \frac { 171 } { 220 }$ 
C)  $- \frac { 171 } { 140 }$ 
D)  $\frac { 3 } { 20 }$ 
A)  $- \frac { 171 } { 221 }$ 
B)  $- \frac { 171 } { 220 }$ 
C)  $- \frac { 171 } { 140 }$ 
D)  $\frac { 3 } { 20 }$

Question 14

Write the word or phrase that best completes each statement or answers the question.
-On the given graph of $y = \cot x$ sketch the graph of $y = \cot ^ { - 1 } x$ as defined in the text. Give the domain and the range.

Accepted Answer

The answer of Write the word or phrase that best...

Question 15

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

Accepted Answer

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

Question 16

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 17

Use Identities to find the exact value.
-$\cos 165 ^ { \circ }$

Accepted Answer

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

Question 18

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

Accepted Answer

A)  $135 ^ { \circ }$ 
B)  $120 ^ { \circ }$ 
C)  $45 ^ { \circ }$ 
D)  $60 ^ { \circ }$ 
A)  $135 ^ { \circ }$ 
B)  $120 ^ { \circ }$ 
C)  $45 ^ { \circ }$ 
D)  $60 ^ { \circ }$

Question 19

Give the degree measure of θ.
-$$\theta = \csc ^ { - 1 } ( - 2 )$$

Accepted Answer

A)  $- 30 ^ { \circ }$ 
B)  $60 ^ { \circ }$ 
C)  $420 ^ { \circ }$ 
D)  $30 ^ { \circ }$ 
A)  $- 30 ^ { \circ }$ 
B)  $60 ^ { \circ }$ 
C)  $420 ^ { \circ }$ 
D)  $30 ^ { \circ }$

Question 20

Decide whether the expression is or is not an identity.
-$$\sqrt { \sec ^ { 2 } x - \tan ^ { 2 } x } = 0$$

Accepted Answer

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

Solve the equation for solutions in the interval [0, 2 π). - $\sin 4 x = \frac { \sqrt { 3 } } { 2 }$

Solve the equation for x, where x is restricted to the given interval. - $y = \cos ( x + 12 ) \text {, for } x \text { in } [ - 12 , \pi - 12 ]$

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. -sec (arctan 4.839)

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

Use the fundamental identities to find the value of the trigonometric function. -Find $\sin \theta$ if $\cos \theta = \frac { 5 } { 6 }$ and $\theta$ is in quadrant IV.

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

Use a sum or difference identity to find the exact value. - $\sin \frac { 7 \pi } { 12 }$

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - $\sqrt { 3 } \sec 2 \theta = 2$

Write the product as a sum or difference of trigonometric functions. - $4 \cos 16 ^ { \circ } \cos 8 ^ { \circ }$

Give the exact value of the expression. - $\sin ( \arctan 2 )$

Solve the equation for x, where x is restricted to the given interval. - $y = 8 \cos 3 x$ , for $x$ in $\left[ 0 , \frac { \pi } { 3 } \right]$

Verify that the equation is an identity. - $\sin \left( x + \frac { \pi } { 2 } \right) = \cos x$

Find the exact value of the expression using the provided information. -Find $\tan ( \mathrm { s } + \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 12 } { 13 }$ , with $\mathrm { s }$ in quadrant II, and $\sin t = \frac { 8 } { 17 }$ , with $t$ in quadrant II.

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

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

Use Identities to find the exact value. - $\cos 165 ^ { \circ }$

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

Give the degree measure of θ. - $\theta = \csc ^ { - 1 } ( - 2 )$

Decide whether the expression is or is not an identity. - $\sqrt { \sec ^ { 2 } x - \tan ^ { 2 } x } = 0$

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

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Exam 7: Trigonometric Identities and Equations

Solve the equation for solutions in the interval [0, 2 π). - sin⁡4x=32\sin 4 x = \frac { \sqrt { 3 } } { 2 }sin4x=23​​

Solve the equation for x, where x is restricted to the given interval. - y=cos⁡(x+12), for x in [−12,π−12]y = \cos ( x + 12 ) \text {, for } x \text { in } [ - 12 , \pi - 12 ]y=cos(x+12), for x in [−12,π−12]

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. -sec (arctan 4.839)

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

Use the fundamental identities to find the value of the trigonometric function. -Find sin⁡θ\sin \thetasinθ if cos⁡θ=56\cos \theta = \frac { 5 } { 6 }cosθ=65​ and θ\thetaθ is in quadrant IV.

Use a sum or difference identity to find the exact value. - sin⁡7π12\sin \frac { 7 \pi } { 12 }sin127π​

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - 3sec⁡2θ=2\sqrt { 3 } \sec 2 \theta = 23​sec2θ=2

Write the product as a sum or difference of trigonometric functions. - 4cos⁡16∘cos⁡8∘4 \cos 16 ^ { \circ } \cos 8 ^ { \circ }4cos16∘cos8∘

Give the exact value of the expression. - sin⁡(arctan⁡2)\sin ( \arctan 2 )sin(arctan2)

Solve the equation for x, where x is restricted to the given interval. - y=8cos⁡3xy = 8 \cos 3 xy=8cos3x , for xxx in [0,π3]\left[ 0 , \frac { \pi } { 3 } \right][0,3π​]

Verify that the equation is an identity. - sin⁡(x+π2)=cos⁡x\sin \left( x + \frac { \pi } { 2 } \right) = \cos xsin(x+2π​)=cosx

Write the word or phrase that best completes each statement or answers the question. -On the given graph of y=cot⁡xy = \cot xy=cotx sketch the graph of y=cot⁡−1xy = \cot ^ { - 1 } xy=cot−1x as defined in the text. Give the domain and the range.

Use the cofunction identities to find an angle θ 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∘)

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

Use Identities to find the exact value. - cos⁡165∘\cos 165 ^ { \circ }cos165∘

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

Give the degree measure of θ. - θ=csc⁡−1(−2)\theta = \csc ^ { - 1 } ( - 2 )θ=csc−1(−2)

Decide whether the expression is or is not an identity. - sec⁡2x−tan⁡2x=0\sqrt { \sec ^ { 2 } x - \tan ^ { 2 } x } = 0sec2x−tan2x​=0