Deck 26: Trigonometric Functions of Any Angle

Question 1

The point is on the terminal side of an angle in standard position.Determine the exact values of the three trigonometric functions sine,cosine,and tangent of the angle θ.​ $( 5,12 )$ ​

A)sin θ = $\frac { 12 } { 13 }$ cos θ = $\frac { 5 } { 13 }$
Tan θ = $\frac { 12 } { 5 }$
B)sin θ = $\frac { 12 } { 13 }$ cos θ = $\frac { 5 } { 13 }$
Tan θ = $\frac { 12 } { 13 }$
C)sin θ = $\frac { 12 } { 5 }$ cos θ = $\frac { 5 } { 13 }$
Tan θ = $\frac { 12 } { 13 }$
D)sin θ = $\frac { 5 } { 13 }$ cos θ = $\frac { 12 } { 13 }$
Tan θ = $\frac { 12 } { 5 }$
E)sin θ = $\frac { 12 } { 13 }$ cos θ = $\frac { 12 } { 5 }$
Tan θ = $\frac { 5 } { 13 }$

Answer 1

sin θ = $\frac { 12 } { 13 }$ cos θ = $\frac { 5 } { 13 }$
Tan θ = $\frac { 12 } { 5 }$

Question 2

Find the reference angle $\theta ^ { \prime }$ ,and sketch $\theta$ and $\theta ^ { \prime }$ in standard position.​ $\theta = \frac { 8 \pi } { 7 }$ ​

A) $\theta ^ { \prime } = \frac { \pi } { 8 }$
B) $\theta ^ { \prime } = \frac { 7 \pi } { 8 }$
C) $\theta ^ { \prime } = 8 \pi$
D) $\theta ^ { \prime } = \frac { \pi } { 7 }$
E) $\theta ^ { \prime } = 7 \pi$

Answer 2

$\theta ^ { \prime } = \frac { \pi } { 7 }$

Question 3

The displacement from equilibrium of an oscillating weight suspended by a spring is given by​ $$y = 2 \cos 6 t$$ ,where y is the displacement in centimeters and t is the time in seconds (see figure).Find the displacement when $$t = \frac { 1 } { 8 }$$ .(Round your answers to two decimal places. )​ ​ ​

A)1.76 cm
B)1.66 cm
C)1.46 cm
D)1.86 cm
E)1.96 cm

Answer 3

1.46 cm

Question 4

Find the reference angle $\theta^\prime$ ,and $\theta$ sketch and $\theta^\prime$ in standard position.​ $\theta = 160 ^ { \circ }$ ​

A) $\theta ^ { \prime } = 10 ^ { \circ }$
B) $\theta^\prime = 30 ^ { \circ }$
C) $\theta ^ { \prime } = - 20 ^ { \circ }$
D) $\theta ^ { \prime } = 20 ^ { \circ }$
E) $\theta ^ { \prime } = 40 ^ { \circ }$

Question 5

Determine the exact values of the three trigonometric functions sine,cosine,and tangent of the angle θ.​ ​ where $a = 12 \text { and } b = 5$ ​

A)sin θ = - $\frac { 5 } { 13 }$ cos θ = - $$\frac { 12 } { 13 }$$
Tan θ = $\frac { 5 } { 12 }$
B)sin θ = - $\frac { 5 } { 12 }$ cos θ = - $$\frac { 12 } { 13 }$$
Tan θ = - $\frac { 5 } { 13 }$
C)sin θ = - $\frac { 12 } { 13 }$ cos θ = - $\frac { 5 } { 13 }$
Tan θ = - $\frac { 5 } { 12 }$
D)sin θ = - $\frac { 5 } { 13 }$ cos θ = - $\frac { 5 } { 12 }$
Tan θ = - $\frac { 12 } { 13 }$
E)sin θ = - $\frac { 5 } { 13 }$ cos θ = - $\frac { 12 } { 13 }$
Tan θ = - $\frac { 5 } { 13 }$

Question 6

Evaluate the trigonometric function.Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode. ) ​
Tan 4.9
​

A)-4.9675
B)-4.8675
C)-5.1675
D)-4.7675
E)-5.2675

Question 7

Evaluate the trigonometric function.Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode. ) ​
Sin 11°
​

A)0.4908
B)0.1908
C)0.5908
D)0.2908
E)0.6908

Question 8

Evaluate the trigonometric function.Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode. ) ​
Tan 306°
​

A)-1.0764
B)-1.2764
C)-1.3764
D)-0.8764
E)-0.9764

Question 9

Evaluate the trigonometric function.Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode. ) ​
Sec 0.21
​

A)1.5225
B)1.1225
C)1.4225
D)1.0225
E)1.3225

Question 10

Find the reference angle $\theta ^ { \prime }$ .​ $\theta = 11.8$ ​

A) $$\theta ^ { \prime } = 2 \pi + 11.8$$ ​
B) $\theta ^ { \prime } = 2 \pi - 11.8$
C) $$\theta ^ { \prime } = 4 \pi - 11.8$$
D) $\theta ^ { \prime } = 4 \pi + 11.8$
E) $\theta ^ { \prime } = 11.8 - 2 \pi$

Question 11

Evaluate the trigonometric function.Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode. )​ $\cos ( - 1180 )$ ​

A)-0.1736
B)0.3264
C)-0.0736
D)0.1264
E)0.2264

Question 12

Determine the exact values of the three trigonometric functions csc,sec,and cot of the angle θ. ​
​​ where $a = 5 \text { and } b = 12$ ​

A)csc θ = $\frac { 13 } { 5 }$ sec θ = $\frac { 13 } { 12 }$
Cot θ = $\frac { 5 } { 12 }$
B)csc θ = $\frac { 13 } { 12 }$ sec θ = $\frac { 13 } { 5 }$
Cot θ = $\frac { 13 } { 12 }$
C)csc θ = $\frac { 13 } { 12 }$ sec θ = $\frac { 13 } { 5 }$
Cot θ = $\frac { 5 } { 12 }$
D)csc θ = $\frac { 5 } { 12 }$ sec θ = $\frac { 13 } { 5 }$
Cot θ = $\frac { 13 } { 12 }$
E)csc θ = $\frac { 13 } { 12 }$ sec θ = $\frac { 5 } { 12 }$
Cot θ = $\frac { 13 } { 5 }$

Question 13

Evaluate the trigonometric function.Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode. ) ​
Sec 221°
​

A)-1.3250
B)-1.0250
C)-0.8250
D)-0.9250
E)-1.2250

Question 14

Find the reference angle $\theta ^ { \prime }$ .​ $$\theta = 5.3$$ ​

A) $\theta ^ { \prime } = \pi - 5.3$
B) $$\theta ^ { \prime } = 5.3 - 2 \pi$$
C) $\theta ^ { \prime } = \pi + 5.3$
D) $\theta ^ { \prime } = 2 \pi - 5.3$
E) $\theta ^ { \prime } = 2 \pi + 5.3$

Question 15

Determine the exact values of the three trigonometric functions csc,sec,and cot of the angle θ.​ ​ where a = 15 and b = 8.
​

A)csc θ = - $$\frac { 17 } { 8 }$$ sec θ = - $\frac { 17 } { 15 }$
Cot θ = $\frac { 15 } { 8 }$
B)cscθ = - $\frac { 17 } { 8 }$ sec θ = - $\frac { 15 } { 8 }$
Cot θ = - $\frac { 17 } { 15 }$
C)csc θ = - $\frac { 17 } { 8 }$ sec θ = - $\frac { 17 } { 15 }$
Cot θ = - $\frac { 17 } { 8 }$
D)csc θ = - $\frac { 15 } { 8 }$ sec θ = - $\frac { 17 } { 15 }$
Cot θ = - $\frac { 17 } { 8 }$
E)csc θ = - $\frac { 17 } { 15 }$ sec θ = - $\frac { 17 } { 8 }$
Cot θ = - $\frac { 15 } { 8 }$

Question 16

Determine the exact values of the three trigonometric functions sine,cosine,and tangent of the angle θ.​ ​ where $a = 5 \text { and } b = 12$ ​

A)sin θ = $\frac { 12 } { 13 }$ cos θ = $\frac { 5 } { 13 }$
Tan θ = $\frac { 12 } { 5 }$
B)sin θ = $\frac { 5 } { 13 }$ cos θ = $\frac { 12 } { 13 }$
Tan θ = $\frac { 12 } { 5 }$
C)sin θ = $\frac { 12 } { 5 }$ cos θ = $\frac { 5 } { 13 }$
Tan θ = $\frac { 12 } { 13 }$
D)sin θ = $\frac { 12 } { 13 }$ cos θ = $\frac { 12 } { 5 }$
Tan θ = $\frac { 5 } { 13 }$
E)sin θ = $\frac { 12 } { 13 }$ cos θ = $\frac { 5 } { 13 }$
Tan θ = $\frac { 12 } { 13 }$

Question 17

Find the reference angle $\theta^\prime$ ,and sketch $\theta$ and $\theta^\prime$ in standard position.​ $\theta = - 135 ^ { \circ }$ ​

A) $\theta ^ { \prime } = - 45 ^ { \circ }$
B) $\theta^\prime = 45 ^ { \circ }$
C) $\theta ^ { \prime }$ = 35 ${}^\circ$
D) $\theta ^ { \prime }$ = 65 ${}^\circ$
E) $\theta ^ { \prime }$ = 55 ${}^\circ$

Question 18

Find the values of the trigonometric functions sine and sec of θ with the given constraint. ​
Function Value
Constra int
Tan θ = - $\frac { 4 } { 3 }$ sin θ > 0
​​

A)sin θ = - $\frac { 4 } { 5 }$ sec θ = $\frac { 5 } { 3 }$
B)sin θ = $\frac { 4 } { 5 }$ sec θ = - $\frac { 5 } { 3 }$
C)sin θ = $\frac { 5 } { 3 }$ sec θ = - $\frac { 4 } { 5 }$
D)sin θ = $\frac { 4 } { 5 }$ sec θ = $\frac { 5 } { 3 }$
E)sin θ = - $\frac { 4 } { 5 }$ sec θ = - $\frac { 5 } { 3 }$

Question 19

The point is on the terminal side of an angle in standard position.Determine the exact values of the three trigonometric functions sine,cosine,and tangent of the angle θ.​ $( 15,8 )$ ​

A)sin θ = $\frac { 8 } { 17 }$ cos θ = $\frac { 8 } { 15 }$
Tan θ = $\frac { 15 } { 17 }$
B)sin θ = $\frac { 8 } { 15 }$ cos θ = $\frac { 15 } { 17 }$
Tan θ = $\frac { 8 } { 17 }$
C)sin θ = $\frac { 15 } { 17 }$ cos θ = $\frac { 8 } { 17 }$
Tan θ = $\frac { 8 } { 15 }$
D)sin θ = $\frac { 8 } { 17 }$ cos θ = $\frac { 15 } { 17 }$
Tan θ = $\frac { 8 } { 15 }$
E)sin θ = $\frac { 8 } { 17 }$ cos θ = $\frac { 15 } { 17 }$
Tan θ = $\frac { 8 } { 17 }$

Question 20

The displacement from equilibrium of an oscillating weight suspended by a spring is given by​ $y = 2 \cos 6 t$ ​ where y is the displacement in centimeters and t is the time in seconds (see figure).Find the displacement when $t = 7$ .(Round your answers to nearest tenth. )​ ​

A)1.200 cm
B)1.100 cm
C)1.000 cm
D)0.800 cm
E)1.300 cm

Question 21

The displacement from equilibrium of an oscillating weight suspended by a spring and subject to the damping effect of friction is given by $y = 2 e ^ { - t } \cos 6 t$ where y is the displacement in centimeters and t is the time in seconds (see figure).Find the displacement when $t = \frac { 1 } { 5 }$ .Round your answer to two decimal places.​ ​ ​

A)0.89 cm
B)0.59 cm
C)0.99 cm
D)1.09 cm
E)0.79 cm

Question 22

Given the figure below,determine the value of $\sin \theta$ .​ ​

A)​ $\sin \theta =$ $- \frac { 12 } { 13 }$
B) $\sin \theta =$ $\frac { 12 } { 13 }$
​
C) $\sin \theta =$ $\frac { 1 } { 2 }$
D) $\sin \theta =$ $- \frac { 1 } { 2 }$
E) $\sin \theta =$ $\frac { 3 } { 4 }$

Question 23

The displacement from equilibrium of an oscillating weight suspended by a spring and subject to the damping effect of friction is given by​ $y = 2 e ^ { - t } \cos 6 t$ where y is the displacement in centimeters and t is the time in seconds (see figure).Find the displacement when $t = \frac { 1 } { 6 }$ .(Round your answer two decimal places. ) ​​ ​
​

A)0.71 cm
B)1.11 cm
C)1.31 cm
D)-0.91 cm
E)0.91 cm

Question 24

Evaluate the trigonometric function.Round your answer to four decimal places.​ $$\tan \left( \frac { \pi } { 3 } \right)$$ ​

A)2.2321
B)1.7321
C)2.7321
D)3.7321
E)4.7321

Question 25

Evaluate the trigonometric function.Round your answer to four decimal places.​ $$\tan \left( - \frac { \pi } { 8 } \right)$$ ​

A)0.5858
B)-0.4142
C)0.0858
D)2.5858
E)1.5858

Question 26

State the quadrant in which θ lies if $\cot \theta$ < 0 and $\sec \theta$ > 0​.

A)Quadrant II
B)Quadrant IV
C)Quadrant I
D)Quadrant III

Question 27

Evaluate the trigonometric function.Round your answer to four decimal places. ​
Csc(-310°)
​

A)2.3054
B)3.3054
C)1.3054
D)4.3054
E)1.8054

Question 28

An airplane,flying at an altitude of 8 miles,is on a flight path that passes directly over an observer (see figure).If $\theta$ is the angle of elevation from the observer to the plane,find the distance from the observer to the plane when $\theta = 60$ °.(Round your answers to two decimal place. )​ ​

A)11.24 mi
B)10.24 mi
C)9.24 mi
D)12.24 mi
E)13.24 mi

Question 29

Evaluate the trigonometric function.Round your answer to four decimal places. ​
Cot 3.35
​

A)7.7286
B)5.7286
C)5.2286
D)6.7286
E)4.7286

Question 30

Determine the exact value of the $\text { tangent }$ of the quadrant angle π.

A)0
B)undefined
C) $- \frac { \sqrt { 3 } } { 2 }$
D) $\frac { 1 } { 2 }$

Question 31

Evaluate the trigonometric function.Round your answer to four decimal places.​ $$\csc \left( - \frac { 13 \pi } { 12 } \right)$$ ​

A)1.8637
B)3.8637
C)4.3637
D)6.8637
E)4.8637

Question 32

Determine the exact value of $\csc \theta$ when $\cot \theta = \frac { 20 } { 21 }$ and $\csc \theta$ < 0.

A) $\csc \theta = - \frac { 11 } { 14 }$
B) $\csc \theta = - \frac { 3 } { 2 }$
C) $\csc \theta = - \frac { 58 } { 21 }$
D) $\csc \theta = - \frac { 8 } { 21 }$
E) $\csc \theta = - \frac { 29 } { 21 }$

Question 33

Evaluate the trigonometric function.Round your answer to four decimal places.​ $$\cot \left( - \frac { 13 \pi } { 5 } \right)$$ ​

A)0.8249
B)3.3249
C)0.3249
D)2.3249
E)1.3249

Question 34

Evaluate the trigonometric function.Round your answer to four decimal places. ​
Tan ( -187°)
​

A)0.8772
B)1.8772
C)-0.1228
D)2.8772
E)0.3772

Question 35

Determine the exact value of $\cot \theta$ when $\cos \theta = \frac { 3 } { 5 }$ and $\cot \theta > 0$ .

A) $\cot \theta = \frac { 3 } { 4 }$
B) $\cot \theta = \frac { 4 } { 3 }$
C) $\cot \theta = \frac { 3 } { 2 }$
D) $\cot \theta = \frac { 7 } { 4 }$
E) $\cot \theta = \frac { 2 } { 3 }$

Question 36

The point $( - 7 , - 24 )$ is on the terminal side of an angle in standard position.Determine the exact value of $\cos \theta$ .

A) $\cos \theta = \frac { 7 } { 24 }$
B) $\cos \theta = - \frac { 18 } { 7 }$
C) $\cos \theta = - \frac { 7 } { 25 }$
D) $\cos \theta = \frac { 31 } { 7 }$
E) $\cos \theta = \frac { 24 } { 7 }$

Question 37

The point $( 5,12 )$ is on the terminal side of an angle in standard position.Determine the exact value of $\sec \theta$ .

A) $\sec \theta = \frac { 5 } { 12 }$
B) $\sec \theta = \frac { 13 } { 5 }$
C) $\sec \theta = \frac { 12 } { 5 }$
D) $\sec \theta = - \frac { 5 } { 13 }$
E) $\sec \theta = - \frac { 5 } { 12 }$

Question 38

The terminal side of θ lies on the line $- 15 x - 8 y = 0$ in the fourth quadrant.Find the exact value of $\cos \theta$ .

A) $\cos \theta = \frac { 8 } { 17 }$
B) $\cos \theta = \frac { 8 } { 25 }$
C) $\cos \theta = \frac { 25 } { 52 }$
D) $\cos \theta = \frac { 17 } { 8 }$
E) $\cos \theta = \frac { 25 } { 17 }$

Question 39

The current I (in amperes)when 100 volts is applied to a circuit is given by​ $I = 5 e ^ { - 2 t } \sin t$ ​ where t is the time (in seconds)after the voltage is applied.Approximate the current at
T = 0.8 second after the voltage is applied.(Round your answers to two decimal places. )
​

A)1.12 ampere
B)1.02 ampere
C)0.72 ampere
D)0.82 ampere
E)0.92 ampere

Question 40

Evaluate the trigonometric function.Round your answer to four decimal places. ​
Cot 171°
​

A)-5.3138
B)-4.3138
C)-6.3138
D)-3.3138
E)-5.8138

Question 41

State the quadrant in which θ lies if $\cos \theta$ > 0 and $\sin \theta$ > 0.

A)Quadrant II
B)Quadrant III
C)Quadrant IV
D)Quadrant I

Question 42

Use a calculator to evaluate cot 335 ${}^\circ$ .Round your answer to four decimal places. ​

A)-2.1445
B)-2.6445
C)-0.4470
D)-1.1970
E)-1.7453

Question 43

Determine the exact value of $\sin \left( - 225 ^ { \circ } \right)$ .

A) $\frac { \sqrt { 2 } } { 2 }$
B) $\frac { \sqrt { 5 } } { 2 }$
C) $- \frac { \sqrt { 2 } } { 2 }$
D)1
E)-1

Question 44

The point (-7,-24)is on the terminal side of an angle in standard position.Determine the exact value of $\cot \theta$ .

A) $\cot \theta = - \frac { 25 } { 24 }$
B) $\cot \theta = \frac { 1 } { 25 }$
C) $\cot \theta = \frac { 7 } { 24 }$
D) $\cot \theta = \frac { 31 } { 7 }$
E) $\cot \theta = - \frac { 24 } { 25 }$

Question 45

The point (7,24)is on the terminal side of an angle in standard position.Determine the exact value of cos θ.

A) $\cot \theta = - \frac { 24 } { 7 }$
B) $\cot \theta = \frac { 24 } { 25 }$
C) $\cot \theta = \frac { 25 } { 24 }$
D) $\cot \theta = - \frac { 24 } { 25 }$
E) $\cot \theta = - \frac { 7 } { 24 }$

Question 46

Given the equation below,determine two solutions such that $0 ^ { \circ } < \theta < 360 ^ { \circ }$ . ​cos θ = $- \frac { \sqrt { 3 } } { 2 }$

A)​θ = 225 ${}^\circ$ ,135 ${}^\circ$
B)θ = 135 ${}^\circ$ ,225 ${}^\circ$
C)θ = 150 ${}^\circ$ ,210 ${}^\circ$
D)θ = 300 ${}^\circ$ ,240 ${}^\circ$
E)θ = 30 ${}^\circ$

Question 47

Find the point $( x , y )$ on the unit circle that corresponds to the real number $t = - \frac { \pi } { 6 }$ .Use your results to evaluate cos t. ​

A)cos t = 1
B)cos t = -1
C) $\cos t = - \frac { \sqrt { 3 } } { 2 }$
D) $\cos t = \frac { \sqrt { 3 } } { 2 }$
E) $\cos t = \frac { 1 } { 2 }$

Question 48

The displacement from equilibrium of an oscillating weight suspended by a spring is given by $y ( t ) = 4 \cos ( 8 t )$ ,where y is the displacement in centimeters and t is the time in seconds.Find the displacement when $t = 1.25$ ,rounding your answer to four decimal places. ​

A)2.2693 cm
B)-3.3563 cm
C)-0.7275 cm
D)-6.5364 cm
E)10.0903 cm

Question 49

Use a calculator to evaluate $\cot 3.3 ^ { \circ }$ .Round your answer to four decimal places.

A)17.3432
B)17.8432
C)6.2599
D)13.8154
E)5.5099

Question 50

Determine the exact value of the $\text { tangent }$ of the quadrant angle $\frac { \pi } { 2 }$ .

A) $- \frac { \sqrt { 2 } } { 2 }$
B) $\frac { 1 } { 2 }$
C) $- \frac { \sqrt { 3 } } { 2 }$
D)0
E)undefined

Question 51

Use a calculator to evaluate $\sin 5 ^ { \circ }$ .Round your answer to four decimal places.

A)-0.9589
B)0.0872
C)-0.4128
D)-0.4282
E)-1.7089

Question 52

Find the point (x,y)on the unit circle that corresponds to the real number $\frac { \pi } { 4 }$ .Use your results to evaluate tan t.

A) $\tan t = - \frac { \sqrt { 3 } } { 2 }$
B) $\tan t = 0$
C) $\tan t = \frac { \sqrt { 3 } } { 2 }$
D) $\tan t = 1$
E) $\tan t = \text { undefmed }$

Question 53

Given the equation below,determine two solutions such that $0 < \theta < 2 \pi$ . sec​ θ = $\frac { 2 \sqrt { 3 } } { 3 }$ ​

A) $\theta = \frac { \pi } { 6 }$
B) $\theta = \frac { 11 \pi } { 8 } , \frac { 7 \pi } { 8 }$
C) $\theta = \frac { 2 \pi } { 4 } , \frac { 5 \pi } { 8 }$
D)​ $\theta = \frac { 2 \pi } { 3 } , \frac { \pi } { 3 }$
E) $\theta = \frac { 11 \pi } { 6 } , \frac { \pi } { 6 }$