Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

A)  $x = \arcsin ( y - \pi )$ 
B)  $x = \arcsin y - \pi$ 
C)  $x = \arcsin y + \pi$ 
D)  $x = \arcsin ( y + \pi )$ 
A)  $x = \arcsin ( y - \pi )$ 
B)  $x = \arcsin y - \pi$ 
C)  $x = \arcsin y + \pi$ 
D)  $x = \arcsin ( y + \pi )$

Question 3

Solve the equation for x, where x is restricted to the given interval.
-$y = - 5 \sin 8 x$, for $x$ in $\left[ - \frac { \pi } { 16 } , \frac { \pi } { 16 } \right]$

Accepted Answer

A)  $x = \frac { 1 } { 5 } \arcsin \left( - \frac { y } { 8 } \right)$ 
B)  $x = - \frac { 1 } { 5 } \arcsin \frac { y } { 8 }$ 
C)  $x = \frac { 1 } { 8 } \arcsin \left( - \frac { y } { 5 } \right)$ 
D)  $x = \frac { 1 } { 8 } \arcsin \frac { y } { 5 }$ 
A)  $x = \frac { 1 } { 5 } \arcsin \left( - \frac { y } { 8 } \right)$ 
B)  $x = - \frac { 1 } { 5 } \arcsin \frac { y } { 8 }$ 
C)  $x = \frac { 1 } { 8 } \arcsin \left( - \frac { y } { 5 } \right)$ 
D)  $x = \frac { 1 } { 8 } \arcsin \frac { y } { 5 }$

Question 4

Solve.
-A rotating beacon is located a distance $\mathrm { d }$ from a long wall. The distance $\mathrm { d }$ is given by $\mathrm { d } = 2 \tan 2 \pi \mathrm { t }$, where $t$ is the time measured in seconds since the beacon started rotating. Solve the equation for $\mathrm { t }$.

Accepted Answer

A)  $t = 2 \pi \tan \frac { d } { 2 }$ 
B)  $t = \frac { 1 } { 2 \pi } \arctan \frac { \mathrm { d } } { 2 }$ 
C)  $t = \frac { 1 } { 2 \pi } \tan \frac { d } { 2 }$ 
D)  $t = 2 \pi \arctan \frac { d } { 2 }$ 
A)  $t = 2 \pi \tan \frac { d } { 2 }$ 
B)  $t = \frac { 1 } { 2 \pi } \arctan \frac { \mathrm { d } } { 2 }$ 
C)  $t = \frac { 1 } { 2 \pi } \tan \frac { d } { 2 }$ 
D)  $t = 2 \pi \arctan \frac { d } { 2 }$

Question 5

Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the figure. Round to
one decimal place.
-Explain why you can find $\cos 1.183$ on your calculator, but $\cos ^ { - 1 } 1.183$ results in an error message.

Accepted Answer

The answer of Use the parallelogram rule to find the...

Question 6

Solve the equation for exact solutions.
-$\arcsin 2 x + 2 \arccos x = \pi$

Accepted Answer

A)  $\left\{ - \frac { \sqrt { 3 } } { 2 } , \frac { \sqrt { 3 } } { 2 } \right\}$ 
B)  $\{ 0 \}$ 
C)  $\left\{ - \frac { \sqrt { 3 } } { 4 } , \frac { \sqrt { 3 } } { 4 } \right\}$ 
D)  $\{ 1 \}$ 
A)  $\left\{ - \frac { \sqrt { 3 } } { 2 } , \frac { \sqrt { 3 } } { 2 } \right\}$ 
B)  $\{ 0 \}$ 
C)  $\left\{ - \frac { \sqrt { 3 } } { 4 } , \frac { \sqrt { 3 } } { 4 } \right\}$ 
D)  $\{ 1 \}$

Question 7

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree.
-sin 2ϴ + cos 2ϴ = 1

Accepted Answer

A) {0°, 90°, 180°, 270°} 
B) {0°, 45°, 180°, 225°} 
C) {0°} 
D) {33°, 57°, 123°, 147°, 213°, 237°, 303°} 
A) {0°, 90°, 180°, 270°} 
B) {0°, 45°, 180°, 225°} 
C) {0°} 
D) {33°, 57°, 123°, 147°, 213°, 237°, 303°}

Question 8

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

Question 9

Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the figure. Round to
one decimal place.
-The equation $\tan \frac { x } { 2 } + \sec \frac { x } { 2 } + 1 = 0$ has no solution in the interval $[ 0,2 \pi )$. Explain what this tells you about the graph of $y = \tan \frac { x } { 2 } + \sec \frac { x } { 2 } + 1$.

Accepted Answer

The graph does not c

Question 10

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

Question 11

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi ) .$$
-$\sqrt { 2 } \cos 2 x = 1$

Accepted Answer

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

Question 12

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree.
-$$\sin 2 \theta = - \frac { 1 } { 2 }$$

Accepted Answer

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

Question 13

Use a calculator to give the value to the nearest degree.
-$$\theta = \sin ^ { - 1 } ( 0.8830 )$$

Accepted Answer

A)  $60 ^ { \circ }$ 
B)  $62 ^ { \circ }$ 
C)  $118 ^ { \circ }$ 
D)  $57 ^ { \circ }$ 
A)  $60 ^ { \circ }$ 
B)  $62 ^ { \circ }$ 
C)  $118 ^ { \circ }$ 
D)  $57 ^ { \circ }$

Question 14

Graph the inverse circular function.
-$$y = \frac { 3 } { 4 } \operatorname { arccot } x$$

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary.
-$\sin ^ { 2 } \theta - \sin \theta - 12 = 0$

Accepted Answer

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

Question 17

Write the following as an algebraic expression in u, u > 0.
-$\sin \left( \arctan \frac { \mathrm { u } } { \sqrt { 5 } } \right)$

Accepted Answer

A)  $\frac { \sqrt { u ^ { 2 } + 5 } } { u ^ { 2 } + 5 }$ 
B)  $u \sqrt { u ^ { 2 } + 5 }$ 
C)  $\frac { u \sqrt { u ^ { 2 } + 5 } } { u ^ { 2 } + 5 }$ 
D)  $\frac { \mathrm { u } \sqrt { \mathrm { u } ^ { 2 } - 5 } } { \mathrm { u } ^ { 2 } - 5 }$ 
A)  $\frac { \sqrt { u ^ { 2 } + 5 } } { u ^ { 2 } + 5 }$ 
B)  $u \sqrt { u ^ { 2 } + 5 }$ 
C)  $\frac { u \sqrt { u ^ { 2 } + 5 } } { u ^ { 2 } + 5 }$ 
D)  $\frac { \mathrm { u } \sqrt { \mathrm { u } ^ { 2 } - 5 } } { \mathrm { u } ^ { 2 } - 5 }$

Question 18

Solve.
-In an electrical circuit, let $\mathrm { V }$ represent the electromagnetic force in volts at $\mathrm { t }$ seconds. Assume $\mathrm { V } = \cos 2 \pi \mathrm { t }$. Find the smallest positive value of $\mathrm { t }$ where $0 \leq \mathrm { t } \leq \frac { 1 } { 2 }$ for $\mathrm { V } = \frac { \sqrt { 2 } } { 2 }$.

Accepted Answer

A)  $\frac { 1 } { 4 } \mathrm { sec }$ 
B)  $\frac { 1 } { 2 } \mathrm { sec }$ 
C)  $\frac { 1 } { 8 } \mathrm { sec }$ 
D)  $0 \mathrm { sec }$ 
A)  $\frac { 1 } { 4 } \mathrm { sec }$ 
B)  $\frac { 1 } { 2 } \mathrm { sec }$ 
C)  $\frac { 1 } { 8 } \mathrm { sec }$ 
D)  $0 \mathrm { sec }$

Question 19

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary.
-$2 \cos 2 \theta + 7 \sin \theta = 5$

Accepted Answer

A)  $\left\{ 30 ^ { \circ } , 330 ^ { \circ } \right\}$ 
B)  $\left\{ 90 ^ { \circ } , 48.6 ^ { \circ } , 131.4 ^ { \circ } \right\}$ 
C)  $\left\{ 30 ^ { \circ } , 210 ^ { \circ } \right\}$ 
D)  $\varnothing$ 
A)  $\left\{ 30 ^ { \circ } , 330 ^ { \circ } \right\}$ 
B)  $\left\{ 90 ^ { \circ } , 48.6 ^ { \circ } , 131.4 ^ { \circ } \right\}$ 
C)  $\left\{ 30 ^ { \circ } , 210 ^ { \circ } \right\}$ 
D)  $\varnothing$

Question 20

Find the exact value of the real number y.
-$y = \arctan ( 1 )$

Accepted Answer

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

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

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

Solve the equation for x, where x is restricted to the given interval. - $y = - 5 \sin 8 x$ , for $x$ in $\left[ - \frac { \pi } { 16 } , \frac { \pi } { 16 } \right]$

Solve. -A rotating beacon is located a distance $\mathrm { d }$ from a long wall. The distance $\mathrm { d }$ is given by $\mathrm { d } = 2 \tan 2 \pi \mathrm { t }$ , where $t$ is the time measured in seconds since the beacon started rotating. Solve the equation for $\mathrm { t }$ .

Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the figure. Round to one decimal place. -Explain why you can find $\cos 1.183$ on your calculator, but $\cos ^ { - 1 } 1.183$ results in an error message.

Solve the equation for exact solutions. - $\arcsin 2 x + 2 \arccos x = \pi$

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. -sin 2ϴ + cos 2ϴ = 1

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

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

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi ) .$ - $\sqrt { 2 } \cos 2 x = 1$

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - $\sin 2 \theta = - \frac { 1 } { 2 }$

Use a calculator to give the value to the nearest degree. - $\theta = \sin ^ { - 1 } ( 0.8830 )$

Graph the inverse circular function. - $y = \frac { 3 } { 4 } \operatorname { arccot } x$ $Graph the inverse circular function. - y = \frac { 3 } { 4 } \operatorname { arccot } x$

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

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary. - $\sin ^ { 2 } \theta - \sin \theta - 12 = 0$

Write the following as an algebraic expression in u, u > 0. - $\sin \left( \arctan \frac { \mathrm { u } } { \sqrt { 5 } } \right)$

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary. - $2 \cos 2 \theta + 7 \sin \theta = 5$

Find the exact value of the real number y. - $y = \arctan ( 1 )$

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 for exact solutions over the interval [0, 2 [0,2π).[ 0,2 \pi ) .[0,2π). - sin⁡2x+sin⁡x=0\sin ^ { 2 } x + \sin x = 0sin2x+sinx=0

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

Solve the equation for x, where x is restricted to the given interval. - y=−5sin⁡8xy = - 5 \sin 8 xy=−5sin8x , for xxx in [−π16,π16]\left[ - \frac { \pi } { 16 } , \frac { \pi } { 16 } \right][−16π​,16π​]

Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the figure. Round to one decimal place. -Explain why you can find cos⁡1.183\cos 1.183cos1.183 on your calculator, but cos⁡−11.183\cos ^ { - 1 } 1.183cos−11.183 results in an error message.

Solve the equation for exact solutions. - arcsin⁡2x+2arccos⁡x=π\arcsin 2 x + 2 \arccos x = \piarcsin2x+2arccosx=π

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. -sin 2ϴ + cos 2ϴ = 1

Solve the equation for solutions in the interval [0, 2 [0,2π)[ 0,2 \pi )[0,2π) - csc⁡3x=0\csc 3 x = 0csc3x=0

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

Solve the equation for solutions in the interval [0, 2 [0,2π).[ 0,2 \pi ) .[0,2π). - 2cos⁡2x=1\sqrt { 2 } \cos 2 x = 12​cos2x=1

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - sin⁡2θ=−12\sin 2 \theta = - \frac { 1 } { 2 }sin2θ=−21​

Use a calculator to give the value to the nearest degree. - θ=sin⁡−1(0.8830)\theta = \sin ^ { - 1 } ( 0.8830 )θ=sin−1(0.8830)

Graph the inverse circular function. - y=34arccot⁡xy = \frac { 3 } { 4 } \operatorname { arccot } xy=43​arccotx

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

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary. - sin⁡2θ−sin⁡θ−12=0\sin ^ { 2 } \theta - \sin \theta - 12 = 0sin2θ−sinθ−12=0

Write the following as an algebraic expression in u, u > 0. - sin⁡(arctan⁡u5)\sin \left( \arctan \frac { \mathrm { u } } { \sqrt { 5 } } \right)sin(arctan5​u​)

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary. - 2cos⁡2θ+7sin⁡θ=52 \cos 2 \theta + 7 \sin \theta = 52cos2θ+7sinθ=5

Find the exact value of the real number y. - y=arctan⁡(1)y = \arctan ( 1 )y=arctan(1)