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)  $\left\{ 0 , \pi , \frac { \pi } { 3 } , \frac { 5 \pi } { 3 } \right\}$

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)  $x = \arcsin ( y + \pi )$

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)  $x = \frac { 1 } { 8 } \arcsin \frac { y } { 5 }$

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)  $t = 2 \pi \arctan \frac { d } { 2 }$

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)  $\{ 1 \}$

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°}

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)  $\varnothing$

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

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)  $\left\{ 15 ^ { \circ } , 165 ^ { \circ } , 195 ^ { \circ } , 345 ^ { \circ } \right\}$

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)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$

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)  $\left\{ 105 ^ { \circ } , 165 ^ { \circ } , 285 ^ { \circ } , 345 ^ { \circ } \right\}$

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)  $57 ^ { \circ }$

Question 14

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

Accepted Answer

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)  $\frac { u \sqrt { u ^ { 2 } + 2 } } { u ^ { 2 } + 2 }$

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)  $\left\{ 45 ^ { \circ } \right\}$

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)  $\frac { \mathrm { u } \sqrt { \mathrm { u } ^ { 2 } - 5 } } { \mathrm { u } ^ { 2 } - 5 }$

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)  $0 \mathrm { sec }$

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)  $\varnothing$

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)  $\frac { 3 \pi } { 4 }$

Exam 6: Inverse Circular Functions and Trigonometric Equations

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. - $y = \frac { 3 } { 4 } \operatorname { arccot } x$ $Graph the inverse circular function. - y = \frac { 3 } { 4 } \operatorname { arccot } x A) B) C) D)$

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. - $y = \arctan ( 1 )$

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)