Question 1

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary.
-$4 \sin ^ { 2 } \theta = 3$

Accepted Answer

A)  $\left\{ 240 ^ { \circ } , 300 ^ { \circ } \right\}$ 
B)  $\varnothing$ 
C)  $\left\{ 60 ^ { \circ } , 120 ^ { \circ } \right\}$ 
D)  $\left\{ 60 ^ { \circ } , 120 ^ { \circ } , 240 ^ { \circ } , 300 ^ { \circ } \right\}$ 
A)  $\left\{ 240 ^ { \circ } , 300 ^ { \circ } \right\}$ 
B)  $\varnothing$ 
C)  $\left\{ 60 ^ { \circ } , 120 ^ { \circ } \right\}$ 
D)  $\left\{ 60 ^ { \circ } , 120 ^ { \circ } , 240 ^ { \circ } , 300 ^ { \circ } \right\}$

Question 2

Express the function as a trigonometric function of x.
-$\sin 3 x$

Accepted Answer

A)  $4 \sin x \cos ^ { 3 } x - 4 \sin ^ { 3 } x \cos x$ 
B)  $\cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$ 
C)  $1 - 8 \sin ^ { 2 } x \cos ^ { 2 } x$ 
D)  $3 \sin x - 4 \sin ^ { 3 } x$ 
A)  $4 \sin x \cos ^ { 3 } x - 4 \sin ^ { 3 } x \cos x$ 
B)  $\cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$ 
C)  $1 - 8 \sin ^ { 2 } x \cos ^ { 2 } x$ 
D)  $3 \sin x - 4 \sin ^ { 3 } x$

Question 3

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

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 4

Write in terms of the cofunction of a complementary angle.
-$\tan 46 ^ { \circ }$

Accepted Answer

A)  $\cot 136 ^ { \circ }$ 
B)  $\cot 134 ^ { \circ }$ 
C)  $\cot 44 ^ { \circ }$ 
D)  $\tan 44 ^ { \circ }$ 
A)  $\cot 136 ^ { \circ }$ 
B)  $\cot 134 ^ { \circ }$ 
C)  $\cot 44 ^ { \circ }$ 
D)  $\tan 44 ^ { \circ }$

Question 5

Use a sum or difference identity to find the exact value.
-$\sin \left( - 15 ^ { \circ } \right)$

Accepted Answer

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

Question 6

Solve the problem.
-Snell's Law states that
$$\frac { \sin \theta _ { 1 } } { \sin \theta _ { 2 } } = \frac { v _ { 1 } } { v _ { 2 } }$$
where $v _ { 1 } , v _ { 2 }$ are the speeds at which light travels through two different mediums, and $\theta _ { 1 } , \theta _ { 2 }$ are the angles of incidence and refraction respectively. Find the angle of refraction if $\frac { \mathrm { v } _ { 1 } } { \mathrm { v } _ { 2 } } = 1.8$ and the angle of incidence is $14 ^ { \circ }$. Give your answer in degrees to the nearest hundredth.

Accepted Answer

A)  $25.81 ^ { \circ }$ 
B)  $7.72 ^ { \circ }$ 
C)  $2.57 ^ { \circ }$ 
D)  $3.85 ^ { \circ }$ 
A)  $25.81 ^ { \circ }$ 
B)  $7.72 ^ { \circ }$ 
C)  $2.57 ^ { \circ }$ 
D)  $3.85 ^ { \circ }$

Question 7

Solve the equation for exact solutions.
-$$\cot ^ { - 1 } x + \cot ^ { - 1 } ( 1 + x ) = \frac { \pi } { 4 }$$

Accepted Answer

A)  $\{ - 5 + 3 \sqrt { 2 } \}$ 
B)  $\{ 2 \}$ 
C)  $\left\{ \frac { 1 } { 2 } \right\}$ 
D)  $\left\{ - \frac { 3 } { 4 } , \frac { 3 } { 4 } \right\}$ 
A)  $\{ - 5 + 3 \sqrt { 2 } \}$ 
B)  $\{ 2 \}$ 
C)  $\left\{ \frac { 1 } { 2 } \right\}$ 
D)  $\left\{ - \frac { 3 } { 4 } , \frac { 3 } { 4 } \right\}$

Question 8

Complete the sentence so the result is an identity. Let x be any real number.
-$$ـــــــــ- 1 = \tan ^ { 2 } x$$

Accepted Answer

A)  $\cos ^ { 2 } x$ 
B)  $\sec ^ { 2 } x$ 
C)  $\sin ^ { 2 } x$ 
D)  $\cot ^ { 2 } x$ 
A)  $\cos ^ { 2 } x$ 
B)  $\sec ^ { 2 } x$ 
C)  $\sin ^ { 2 } x$ 
D)  $\cot ^ { 2 } x$

Question 9

Solve the problem.
-A painting 1 meter high and 3 meters from the floor will cut off an angle $\theta$ to an observer, where $\theta = \tan ^ { - 1 } \left( \frac { \mathrm { x } } { \mathrm { x } ^ { 2 } + 1.8 } \right)$, assuming that the observer is $x$ feet from the wall where the painting is displayed and that the eyes of the observer are $1.8$ meters above the ground (see the figure). Find the value of $\theta$ for $x = 2$. Round to the nearest tenth of a degree.

Accepted Answer

A)  $27.8 ^ { \circ }$ 
B)  $19.0 ^ { \circ }$ 
C)  $21.5 ^ { \circ }$ 
D)  $16.5 ^ { \circ }$ 
A)  $27.8 ^ { \circ }$ 
B)  $19.0 ^ { \circ }$ 
C)  $21.5 ^ { \circ }$ 
D)  $16.5 ^ { \circ }$

Question 10

Solve the problem.
-Two voltages, $V _ { 1 } = 70 \sin ( 100 \pi t )$ and $V _ { 2 } = 240 \cos ( 100 \pi t )$, are added in a circuit to produce the sum $V = V _ { 1 } + V _ { 2 }$. Graph $V$ in $[ - 0.02,0.02,0.002 ]$ by $[ - 250,250,20 ]$ and use the graph to estimate values for $\mathrm { a }$ and $\mathrm { k }$ so that $\mathrm { V } = \mathrm { a } \sin ( 100 \pi \mathrm { t } + \mathrm { k } )$. Round $\mathrm { k }$ to the nearest ten-thousandth, if necessary.

Accepted Answer

A)  $\mathrm { a } = 25 ; \mathrm { k } = 1.287$ 
B)  $\mathrm { a } = 250 ; \mathrm { k } = 404.3233$ 
C)  $a = 250 ; k = 1.287$ 
D)  $\mathrm { a } = 250 ; \mathrm { k } = 89.1565$ 
A)  $\mathrm { a } = 25 ; \mathrm { k } = 1.287$ 
B)  $\mathrm { a } = 250 ; \mathrm { k } = 404.3233$ 
C)  $a = 250 ; k = 1.287$ 
D)  $\mathrm { a } = 250 ; \mathrm { k } = 89.1565$

Question 11

Solve the problem.
-Suppose that for an electrical appliance, voltage is given by $\mathrm { V } = 189 \sin 120 \pi \mathrm { t }$ and amperage by $I = 1.07 \sin 120 \pi \mathrm { t }$, where $t$ is time in seconds. Use identities to write the wattage $W = V I$ in the form $\mathrm { W } = \mathrm { a } \cos \omega \mathrm { t } + \mathrm { c }$.

Accepted Answer

A)  $W = - 101.115 \cos 240 \pi t + 101.115$ 
B)  $\mathrm { W } = - 101.115 \cos 120 \pi t + 101.115$ 
C)  $\mathrm { W } = - 202.23 \cos 240 \pi \mathrm { t } + 202.23$ 
D)  $W = 101.115 \cos 240 \pi t - 101.115$ 
A)  $W = - 101.115 \cos 240 \pi t + 101.115$ 
B)  $\mathrm { W } = - 101.115 \cos 120 \pi t + 101.115$ 
C)  $\mathrm { W } = - 202.23 \cos 240 \pi \mathrm { t } + 202.23$ 
D)  $W = 101.115 \cos 240 \pi t - 101.115$

Question 12

Find the exact value of the expression using the provided information.
-Find tan(s - t) given that cos s = - 135 , with s in quadrant II, and sin t = 178 , with t in quadrant II.

Accepted Answer

A)  - 220171 
B)  - 203 
C)  - 140171 
D)  203 
A)  - 220171 
B)  - 203 
C)  - 140171 
D)  203

Question 13

Give the degree measure of θ.
-$$\theta = \arccos ( 0 )$$

Accepted Answer

A)  $0 ^ { \circ }$ 
B)  $180 ^ { \circ }$ 
C)  $90 ^ { \circ }$ 
D)  $45 ^ { \circ }$ 
A)  $0 ^ { \circ }$ 
B)  $180 ^ { \circ }$ 
C)  $90 ^ { \circ }$ 
D)  $45 ^ { \circ }$

Question 14

Write in terms of the cofunction of a complementary angle.
-$\sin 76 ^ { \circ }$

Accepted Answer

A)  $\cos 104 ^ { \circ }$ 
B)  $\csc 14 ^ { \circ }$ 
C)  $\cos 14 ^ { \circ }$ 
D)  $\csc 104 ^ { \circ }$ 
A)  $\cos 104 ^ { \circ }$ 
B)  $\csc 14 ^ { \circ }$ 
C)  $\cos 14 ^ { \circ }$ 
D)  $\csc 104 ^ { \circ }$

Question 15

Use an identity to write the expression as a single trigonometric function or as a single number.
-$$\frac { 2 \tan 15 ^ { \circ } } { 1 - \tan ^ { 2 } 15 ^ { \circ } }$$

Accepted Answer

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

Question 16

Solve the problem.
-A rotating beacon is located a distance $d$ from a long wall. The distance $d$ is given by $\mathrm { d } = 3 \tan 2 \pi \mathrm { t }$,
where $t$ is the time measured in seconds since the beacon started rotating. Solve the equation for $\mathrm { t }$.

Accepted Answer

A)  $\mathrm { t } = \frac { 1 } { 2 \pi } \tan \frac { \mathrm { d } } { 3 }$ 
B)  $t = 2 \pi \arctan \frac { \mathrm { d } } { 3 }$ 
C)  $t = \frac { 1 } { 2 \pi } \arctan \frac { d } { 3 }$ 
D)  $t = 2 \pi \tan \frac { d } { 3 }$ 
A)  $\mathrm { t } = \frac { 1 } { 2 \pi } \tan \frac { \mathrm { d } } { 3 }$ 
B)  $t = 2 \pi \arctan \frac { \mathrm { d } } { 3 }$ 
C)  $t = \frac { 1 } { 2 \pi } \arctan \frac { d } { 3 }$ 
D)  $t = 2 \pi \tan \frac { d } { 3 }$

Question 17

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

Accepted Answer

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

Question 18

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

Accepted Answer

A)  $\sin \theta$ 
B)  $\cos ^ { 3 } \theta$ 
C)  $\sec ^ { 2 } \theta$ 
D)  $\tan ^ { 2 } \theta$ 
A)  $\sin \theta$ 
B)  $\cos ^ { 3 } \theta$ 
C)  $\sec ^ { 2 } \theta$ 
D)  $\tan ^ { 2 } \theta$

Question 19

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

Accepted Answer

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

Question 20

Use the fundamental identities to find the value of the trigonometric function.
-Find $\csc \theta$ if $\cot \theta = - \sqrt { 15 }$ and $\theta$ is in quadrant II.

Accepted Answer

A)  $- 4$ 
B)  $- \frac { 1 } { 4 }$ 
C)  4 
D)  $\frac { 1 } { 4 }$ 
A)  $- 4$ 
B)  $- \frac { 1 } { 4 }$ 
C)  4 
D)  $\frac { 1 } { 4 }$

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary. - $4 \sin ^ { 2 } \theta = 3$

Express the function as a trigonometric function of x. - $\sin 3 x$

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

Write in terms of the cofunction of a complementary angle. - $\tan 46 ^ { \circ }$

Use a sum or difference identity to find the exact value. - $\sin \left( - 15 ^ { \circ } \right)$

Solve the equation for exact solutions. - $\cot ^ { - 1 } x + \cot ^ { - 1 } ( 1 + x ) = \frac { \pi } { 4 }$

Complete the sentence so the result is an identity. Let x be any real number. - $ـــــــــ- 1 = \tan ^ { 2 } x$

Find the exact value of the expression using the provided information. -Find tan(s - t) given that cos s = - 135 , with s in quadrant II, and sin t = 178 , with t in quadrant II.

Give the degree measure of θ. - $\theta = \arccos ( 0 )$

Write in terms of the cofunction of a complementary angle. - $\sin 76 ^ { \circ }$

Use an identity to write the expression as a single trigonometric function or as a single number. - $\frac { 2 \tan 15 ^ { \circ } } { 1 - \tan ^ { 2 } 15 ^ { \circ } }$

Solve the problem. -A rotating beacon is located a distance $d$ from a long wall. The distance $d$ is given by $\mathrm { d } = 3 \tan 2 \pi \mathrm { t }$ , where $t$ is the time measured in seconds since the beacon started rotating. Solve the equation for $\mathrm { t }$ .

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

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

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

Use the fundamental identities to find the value of the trigonometric function. -Find $\csc \theta$ if $\cot \theta = - \sqrt { 15 }$ and $\theta$ is in quadrant II.

Equations and Inequalities

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Exam 7: Trigonometric Identities and Equations

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary. - 4sin⁡2θ=34 \sin ^ { 2 } \theta = 34sin2θ=3

Express the function as a trigonometric function of x. - sin⁡3x\sin 3 xsin3x

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - cos⁡2θ=32\cos 2 \theta = \frac { \sqrt { 3 } } { 2 }cos2θ=23​​

Write in terms of the cofunction of a complementary angle. - tan⁡46∘\tan 46 ^ { \circ }tan46∘

Use a sum or difference identity to find the exact value. - sin⁡(−15∘)\sin \left( - 15 ^ { \circ } \right)sin(−15∘)

Solve the equation for exact solutions. - cot⁡−1x+cot⁡−1(1+x)=π4\cot ^ { - 1 } x + \cot ^ { - 1 } ( 1 + x ) = \frac { \pi } { 4 }cot−1x+cot−1(1+x)=4π​

Complete the sentence so the result is an identity. Let x be any real number. - ـــــــــ−1=tan⁡2xـــــــــ- 1 = \tan ^ { 2 } xـــــــــ−1=tan2x