Question 1

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary.
-$$2 \cos ^ { 3 } \theta = \cos \theta$$

Accepted Answer

A)  $\left\{ 90 ^ { \circ } , 270 ^ { \circ } \right\}$ 
B)  $\left\{ 45 ^ { \circ } , 90 ^ { \circ } , 135 ^ { \circ } , 225 ^ { \circ } , 270 ^ { \circ } , 315 ^ { \circ } \right\}$ 
C)  $\left\{ 45 ^ { \circ } , 135 ^ { \circ } , 225 ^ { \circ } , 315 ^ { \circ } \right\}$ 
D)  $\varnothing$ 
A)  $\left\{ 90 ^ { \circ } , 270 ^ { \circ } \right\}$ 
B)  $\left\{ 45 ^ { \circ } , 90 ^ { \circ } , 135 ^ { \circ } , 225 ^ { \circ } , 270 ^ { \circ } , 315 ^ { \circ } \right\}$ 
C)  $\left\{ 45 ^ { \circ } , 135 ^ { \circ } , 225 ^ { \circ } , 315 ^ { \circ } \right\}$ 
D)  $\varnothing$

Question 2

Give the exact value of the expression.
-$\arccos \left( \cos \frac { 4 \pi } { 3 } \right)$

Accepted Answer

A)  $\frac { \pi } { 6 }$ 
B)  $\frac { \pi } { 3 }$ 
C)  $\frac { 2 \pi } { 3 }$ 
D)  $\frac { 7 \pi } { 6 }$ 
A)  $\frac { \pi } { 6 }$ 
B)  $\frac { \pi } { 3 }$ 
C)  $\frac { 2 \pi } { 3 }$ 
D)  $\frac { 7 \pi } { 6 }$

Question 3

Solve.
-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 $t$.

Accepted Answer

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

Question 4

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$$y = \cos ^ { - 1 } ( 0.8910 )$$

Accepted Answer

A)  $0.5063492$ 
B)  $0.4712533$ 
C)  $0.4365083$ 
D)  $1.1000010$ 
A)  $0.5063492$ 
B)  $0.4712533$ 
C)  $0.4365083$ 
D)  $1.1000010$

Question 5

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

Accepted Answer

A)  $\varnothing$ 
B)  $\left\{ \frac { \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 15 \pi } { 8 } \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
D)  $\left\{ 0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } \right\}$ 
A)  $\varnothing$ 
B)  $\left\{ \frac { \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 15 \pi } { 8 } \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
D)  $\left\{ 0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } \right\}$

Question 6

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi ) .$$
-$\sin 2 x + \sin x = 0$

Accepted Answer

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

Question 7

Give the degree measure of .
-$$\theta = \cos ^ { - 1 } \left( \frac { \sqrt { 3 } } { 2 } \right)$$

Accepted Answer

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

Question 8

Solve the equation (x in radians and 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.
-$\cos ^ { 2 } x - \cos x = 0$

Accepted Answer

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

Question 9

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

Question 10

Provide an appropriate response.
-$$\text { True or false? The statement } \sin ^ { - 1 } ( \sin x ) = x \text { for all real numbers in the interval } - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } \text {. }$$

Accepted Answer

A) True 
 B)False

Question 11

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

Accepted Answer

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

Question 12

Solve the equation (x in radians and 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.
-$2 \sin ^ { 2 } x + \sin x = 1$

Accepted Answer

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

Question 13

Solve the equation for exact solutions.
-$4 \cos ^ { - 1 } x = \pi$

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

Find the exact value of the real number y.
-$$y = \csc ^ { - 1 } \text { (2) }$$

Accepted Answer

A)  $\frac { \pi } { 3 }$ 
B)  $- \frac { \pi } { 6 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $- \frac { \pi } { 3 }$ 
A)  $\frac { \pi } { 3 }$ 
B)  $- \frac { \pi } { 6 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $- \frac { \pi } { 3 }$

Question 17

Solve the problem.
-The area of a triangle is given by $$\mathrm { A } = \frac { 1 } { 2 } \mathrm { ab } \sin \mathrm { C } \text {, }$$
where $a$ and $b$ are the lengths of two of the sides and $C$ is the included angle. If $A = 29$ in. $2 , a = 8$ in., $b = 9$ in., and $C$ is an acute angle, what must $C$ be? Give your answer in degrees to the nearest hundredth.

Accepted Answer

A)  $26.83 ^ { \circ }$ 
B)  $53.66 ^ { \circ }$ 
C)  $126.34 ^ { \circ }$ 
D)  No such triangle exists. 
A)  $26.83 ^ { \circ }$ 
B)  $53.66 ^ { \circ }$ 
C)  $126.34 ^ { \circ }$ 
D)  No such triangle exists.

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

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

Question 20

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree.
-sin 2ϴ = cos ϴ

Accepted Answer

A) {0°, 120°, 180°, 240°} 
B) {105°, 165°, 285°, 345°} 
C) {15°, 165°, 195°, 345°} 
D) {30°, 90°, 150°, 270°} 
A) {0°, 120°, 180°, 240°} 
B) {105°, 165°, 285°, 345°} 
C) {15°, 165°, 195°, 345°} 
D) {30°, 90°, 150°, 270°}

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary. - $2 \cos ^ { 3 } \theta = \cos \theta$

Give the exact value of the expression. - $\arccos \left( \cos \frac { 4 \pi } { 3 } \right)$

Solve. -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 $t$ .

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $y = \cos ^ { - 1 } ( 0.8910 )$

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

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi ) .$ - $\sin 2 x + \sin x = 0$

Give the degree measure of . - $\theta = \cos ^ { - 1 } \left( \frac { \sqrt { 3 } } { 2 } \right)$

Solve the equation (x in radians and 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. - $\cos ^ { 2 } x - \cos x = 0$

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

Provide an appropriate response. - $\text { True or false? The statement } \sin ^ { - 1 } ( \sin x ) = x \text { for all real numbers in the interval } - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } \text {. }$

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

Solve the equation (x in radians and 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. - $2 \sin ^ { 2 } x + \sin x = 1$

Solve the equation for exact solutions. - $4 \cos ^ { - 1 } x = \pi$

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

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

Find the exact value of the real number y. - $y = \csc ^ { - 1 } \text { (2) }$

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

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

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. -sin 2ϴ = cos ϴ

Trigonometric Functions

Acute Angles and Right Triangles

Radian Measure and the Unit Circle

Graphs of the Circular Functions

Trigonometric Identities

Applications of Trigonometry and Vectors

Complex Numbers, Polar Equations, and Parametric Equations

Filters

Exam 6: Inverse Circular Functions and Trigonometric Equations

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary. - 2cos⁡3θ=cos⁡θ2 \cos ^ { 3 } \theta = \cos \theta2cos3θ=cosθ

Give the exact value of the expression. - arccos⁡(cos⁡4π3)\arccos \left( \cos \frac { 4 \pi } { 3 } \right)arccos(cos34π​)

Solve. -In an electric circuit, the electromotive force is defined as V=cos⁡2πt\mathrm { V } = \cos 2 \pi \mathrm { t }V=cos2πt , where t\mathrm { t }t is time in seconds. Solve the equation for ttt .

Use a calculator to give the real number value. Round the answer to 7 decimal places. - y=cos⁡−1(0.8910)y = \cos ^ { - 1 } ( 0.8910 )y=cos−1(0.8910)

Solve the equation for solutions in the interval [0, 2 [0,2π)[ 0,2 \pi )[0,2π) - cos⁡2x=2−cos⁡2x\cos 2 x = \sqrt { 2 } - \cos 2 xcos2x=2​−cos2x

Solve the equation for solutions in the interval [0, 2 [0,2π).[ 0,2 \pi ) .[0,2π). - sin⁡2x+sin⁡x=0\sin 2 x + \sin x = 0sin2x+sinx=0

Give the degree measure of . - θ=cos⁡−1(32)\theta = \cos ^ { - 1 } \left( \frac { \sqrt { 3 } } { 2 } \right)θ=cos−1(23​​)

Solve the equation (x in radians and 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−cos⁡x=0\cos ^ { 2 } x - \cos x = 0cos2x−cosx=0