Deck 8: The Unit Circle and Functions of Trigonometry

Question 1

(a) Convert $-240^{\circ}$ to radians.
(b) Convert $\frac{7 \pi}{15}$ to degrees.

Answer 1

(a) $-\frac{4 \pi}{3}$
(b) $84^{\circ}$

Question 2

A wheel is turning at a speed of 200 revolutions per minute. Through how many degrees does a point on the edge of the wheel move in 1.5 seconds?

Answer 2

$1800^{\circ}$

Question 3

Find the angle of smallest positive measure coterminal with $-418^{\circ}$ .

Answer 3

$302^{\circ}$

Question 4

A central angle of a circle with radius 50 centimeters cuts off an arc of 120 centimeters. Find each measure.
(a) The radian measure of the angle.
(b) The area of the sector with this central angle.

Question 5

Find the length of an arc intercepted by an angle of $60^{\circ}$ in a circle with radius 15 inches.

Question 6

graph the given function over a two-period interval. Identify asymptotes when applicable.

- $y=\tan (x-\pi)$

Question 7

graph the given function over a two-period interval. Identify asymptotes when applicable.

- $y=-\sin 4 x$

Question 8

graph the given function over a two-period interval. Identify asymptotes when applicable.

- $y=2 \sin \left(x+\frac{\pi}{2}\right)+1$

Question 9

graph the given function over a two-period interval. Identify asymptotes when applicable.

-The average monthly low temperature (in $\mathrm{F}^{\circ}$ ) in London can be modeled using the trigonometric function defined by $f(x)=10 \sin \left[\frac{\pi}{6}(x-4.2)\right]+46$ , where $x$ is the month and $x=1$ corresponds to January.
(a) Graph $f$ over the interval $1 \leq x \leq 25$ .
(b) Determine the amplitude, period, phase shift, and vertical translation of $f$ .
(c) What is the average monthly low temperature for the month of October?
(d) Find the maximum and minimum average monthly low temperatures and the months when they occur.
(e) What would be an approximation for the average yearly low temperature in London? How is this related to the vertical translation of the sine function in the formula for $f$ ?

Question 10

If $\cos \theta<0$ and $\tan \theta>0$ , in which quadrant does $\theta$ lie?

Question 11

Use the figure to find the following:

(a) The smallest positive angle coterminal with $\theta$ .
(b) The values of $x$ and $y$ .
(c) The values of the six trigonometric functions of $\theta$ .
(d) The radian measure of $\theta$ .

Question 12

If $(-3,4)$ is on the terminal side of an angle $\theta$ in standard position, find $\sin \theta, \cos \theta$ , and $\tan \theta$ .

Question 13

If $\sin \theta=\frac{3}{5}$ and $\theta$ is in quadrant II find the values of the other trigonometric functions of $\theta$ .

Question 14

Find the exact values of each part labeled with a letter.

Question 15

Find the exact value of $\cot \left(-480^{\circ}\right)$ .

Question 16

Use a calculator to approximate the following.
(a) $\cos 28^{\circ} 52^{\prime}$
(b) $\cot 52.1896^{\circ}$
(c) $\csc 19.635^{\circ}$

Question 17

Use a calculator to approximate $\theta$ in the interval $\left[0,90^{\circ}\right]$ , if $\sin \theta=0.7313537016$ .

Question 18

Solve the triangle.

Question 19

A jet leaves an airport on a bearing of $\mathrm{S} 37^{\circ} \mathrm{E}$ and travels for 324 miles. It then turns and continues on a bearing of $\mathrm{N} 53^{\circ} \mathrm{E}$ for 215 miles. How far is the jet from the airport?

Question 20

To find the height of a fence in a baseball park, a groundskeeper found that the angle of elevation from a point 25.5 meters from the base of the fence is $23^{\circ} 54^{\prime}$ . What is the height of the fence?

Question 21

the formula $s(t)=-5 \cos 3 \pi t$ gives the height (in inches) of a weight attached to a spring after $t$ seconds.
-Find the maximum height that the weight rises above the equilibrium position of $y=0$ .

Question 22

the formula $s(t)=-5 \cos 3 \pi t$ gives the height (in inches) of a weight attached to a spring after $t$ seconds.
-When does the weight reach its maximum height if $t \geq 0$ ?

Question 23

(a) Convert $315^{\circ}$ to radians.
(b) Convert $\frac{11 \pi}{12}$ to degrees.

Question 24

A wheel is turning at a speed of 100 revolutions per minute. Through how many degrees does a point on the edge of the wheel move in 1.05 second?

Question 25

Find the angle of smallest positive measure coterminal with $-634^{\circ}$ .

Question 26

A central angle of a circle with radius 45 centimeters cuts off an arc of 165 centimeters. Find each measure.
(a) The radian measure of the angle.
(b) The area of the sector with this central angle.

Question 27

Find the length of an arc intercepted by an angle of $85^{\circ}$ in a circle with radius 19 inches.
graph the given function over a two-period interval. Identify asymptotes when applicable.

Question 28

graph the given function over a two-period interval. Identify asymptotes when applicable

- $y=\tan (x+\pi)$

Question 29

graph the given function over a two-period interval. Identify asymptotes when applicable

- $y=-\cos 2 x$

Question 30

graph the given function over a two-period interval. Identify asymptotes when applicable

- $y=2 \cos (x+\pi)+1$

Question 31

graph the given function over a two-period interval. Identify asymptotes when applicable

-The average monthly low temperature (in $\mathrm{F}^{\circ}$ ) in Paris can be modeled using the trigonometric function defined by $f(x)=12 \sin \left[\frac{\pi}{6}(x-4.1)\right]+46$ , where $x$ is the month and $x=1$ corresponds to January.
(a) Graph $f$ over the interval $1 \leq x \leq 25$ .
(b) Determine the amplitude, period, phase shift, and vertical translation of $f$ .
(c) What is the average monthly low temperature for the month of October?
(d) Find the maximum and minimum average monthly low temperatures and the months when they occur.
(e) What would be an approximation for the average yearly low temperature in Paris? How is this related to the vertical translation of the sine function in the formula for $f$ ?

Question 32

If $\csc \theta>0$ and $\sec \theta<0$ , in which quadrant does $\theta$ lie?

Question 33

Use the figure to find the following:

(a) The smallest positive angle coterminal with $\theta$ .
(b) The values of $x$ and $y$ .
(c) The values of the six trigonometric functions of $\theta$ .
(d) The radian measure of $\theta$ .

Question 34

If $(2,-6)$ is on the terminal side of angle $\theta$ in standard position, find $\sin \theta, \cos \theta$ , and $\tan \theta$ .

Question 35

If $\tan \theta=\frac{4}{3}$ and $\theta$ is in quadrant III find the values of the other trigonometric functions of $\theta$ .

Question 36

Find the exact values of each part labeled with a letter.

Question 37

Find the exact value of $\sec \left(585^{\circ}\right)$ .

Question 38

Use a calculator to approximate the following.
(a) $\tan 145^{\circ} 23^{\prime}$
(b) $\sin 38.251^{\circ}$
(c) $\sec 18.563^{\circ}$

Question 39

Use a calculator to approximate $\theta$ to the nearest tenth in the interval $\left[0,90^{\circ}\right]$ , if $\cos \theta=0.9086146585$ .

Question 40

Solve the triangle.

Question 41

A jet leaves an airport on a bearing of $\mathrm{N} 47^{\circ} \mathrm{W}$ and travels for 456 miles. It then turns and continues on a bearing of $\mathrm{S} 43^{\circ} \mathrm{W}$ for 381 miles. How far is the jet from the airport?

Question 42

To find the height of a tree, an arborist found that the angle of elevation from a point 52.6 feet from the base of the tree is $43^{\circ} 21^{\prime}$ . What is the height of the tree?

Question 43

the formula $s(t)=-11 \cos 5 \pi t$ gives the height (in inches) of a weight attached to a spring after $t$ seconds.
-Find the maximum height that the weight rises above the equilibrium position of $y=0$ .

Question 44

the formula $s(t)=-11 \cos 5 \pi t$ gives the height (in inches) of a weight attached to a spring after $t$ seconds.
-When does the weight reach its maximum height if $t \geq 0$ ?

Question 45

(a) Convert $-225^{\circ}$ to radians.
(b) Convert $\frac{3 \pi}{10}$ to degrees.

Question 46

A wheel is turning at a speed of 72 revolutions per minute. Through how many degrees does a point on the edge of the wheel move in 0.75 seconds?

Question 47

Find the angle of smallest positive measure coterminal with $-828^{\circ}$ .

Question 48

A central angle of a circle with radius 50 centimeters cuts off an arc of 80 centimeters. Find each measure.
(a) The radian measure of the angle.
(b) The area of the sector with this central angle.

Question 49

Find the length of an arc intercepted by an angle of $120^{\circ}$ in a circle with radius 7 inches.

Question 50

graph the given function over a two-period interval. Identify asymptotes when applicable.

- $y=\tan \left(x-\frac{3 \pi}{2}\right)$

Question 51

graph the given function over a two-period interval. Identify asymptotes when applicable.

- $y=-\sin 2 x$

Question 52

graph the given function over a two-period interval. Identify asymptotes when applicable.

- $y=2 \sin \left(x-\frac{\pi}{2}\right)+1$

Question 53

graph the given function over a two-period interval. Identify asymptotes when applicable.

-The average monthly high temperature $\left(\right.$ in $\mathrm{F}^{\circ}$ ) in London can be modeled using the trigonometric function defined by $f(x)=14 \sin \left[\frac{\pi}{6}(x-4.2)\right]+57$ , where $x$ is the month and $x=1$ corresponds to January.
(a) Graph $f$ over the interval $1 \leq x \leq 25$ .
(b) Determine the amplitude, period, phase shift, and vertical translation of $f$ .
(c) What is the average monthly high temperature for the month of October?
(d) Find the maximum and minimum average monthly high temperatures and the months when they occur.
(e) What would be an approximation for the average yearly high temperature in London? How is this related to the vertical translation of the sine function in the formula for $f$ ?

Question 54

If $\sec \theta<0$ and $\csc \theta>0$ , in which quadrant does $\theta$ lie?

Question 55

Use the figure to find the following:

(a) The smallest positive angle coterminal with $\theta$ ..
(b) The values of $x$ and $y$ .
(c) The values of the six trigonometric functions of $\theta$
(d) The radian measure of $\theta$ .

Question 56

If $(-5,-6)$ is on the terminal side of angle $\theta$ in standard position, find $\sin \theta, \cos \theta$ , and $\tan \theta$ .

Question 57

If $\cos \theta=\frac{5}{13}$ and $\theta$ is in quadrant IV find the values of the other trigonometric functions of $\theta$ .

Question 58

Find the exact values of each part labeled with a letter.

Question 59

Find the exact value of $\csc \left(570^{\circ}\right)$ .

Question 60

Use a calculator to approximate the following.
(a) $\csc 85^{\circ} 14^{\prime}$
(b) $\cos 101.655^{\circ}$
(c) $\tan 221.125^{\circ}$

Question 61

Use a calculator to approximate $\theta$ to the nearest tenth in the interval $\left[0,90^{\circ}\right]$ , if $\cos \theta=0.9086146585$ .

Question 62

Solve the triangle.

Question 63

A jet leaves an airport on a bearing of $\mathrm{S} 29^{\circ} \mathrm{W}$ and travels for 245 miles. It then turns and continues on a bearing of $\mathrm{N} 61^{\circ} \mathrm{W}$ for 506 miles. How far is the jet from the airport?

Question 64

To find the height of a building, a surveyor found that the angle of elevation from a point 42.5 feet from the base of the building is $79^{\circ} 15^{\prime}$ . What is the height of the building?

Question 65

the formula $s(t)=-9 \cos \frac{\pi}{2} t$ gives the height (in inches) of a weight attached to a spring after $t$ seconds.
-Find the maximum height that the weight rises above the equilibrium position of $y=0$ .

Question 66

the formula $s(t)=-9 \cos \frac{\pi}{2} t$ gives the height (in inches) of a weight attached to a spring after $t$ seconds.
-When does the weight reach its maximum height if $t \geq 0$ ?

Question 67

(a) Convert $300^{\circ}$ to radians.
(b) Convert $-\frac{11 \pi}{30}$ to degrees.

Question 68

A wheel is turning at a speed of 72 revolutions per minute. Through how many degrees does a point on the edge of the wheel move in 3 seconds?

Question 69

Find the angle of smallest positive measure coterminal with $-559^{\circ}$ .

Question 70

A central angle of a circle with radius 35 centimeters cuts off an arc of 100 centimeters. Find each measure.
(a) The radian measure of the angle.
(b) The area of the sector with this central angle.

Question 71

Find the length of an arc intercepted by an angle of $235^{\circ}$ in a circle with radius 27 inches.

Question 72

graph the given function over a two-period interval. Identify asymptotes when applicable.

- $y=\tan \left(x+\frac{\pi}{2}\right)$

Question 73

graph the given function over a two-period interval. Identify asymptotes when applicable.

- $y=-\cos 4 x$

Question 74

graph the given function over a two-period interval. Identify asymptotes when applicable.

- $y=3 \cos (x-\pi)+2$

Question 75

graph the given function over a two-period interval. Identify asymptotes when applicable.

-The average monthly high temperature (in $\mathrm{F}^{\circ}$ ) in Paris can be modeled using the trigonometric function defined by $f(x)=16.5 \sin \left[\frac{\pi}{6}(x-4.1)\right]+59.5$ , where $x$ is the month and $x=1$ corresponds to January.
(a) Graph $f$ over the interval $1 \leq x \leq 25$ .
(b) Determine the amplitude, period, phase shift, and vertical translation of $f$ .
(c) What is the average monthly high temperature for the month of October?
(d) Find the maximum and minimum average monthly high temperatures and the months when they occur.
(e) What would be an approximation for the average yearly high temperature in Paris? How is this related to the vertical translation of the sine function in the formula for $f$ ?

Question 76

If $\csc \theta<0$ and $\tan \theta<0$ , in which quadrant does $\theta$ lie?

Question 77

Use the figure to find the following:

(a) The smallest positive angle coterminal with $\theta$ .
(b) The values of $x$ and $y$ .
(c) The values of the six trigonometric functions of $\theta$ .
(d) The radian measure of $\theta$ .

Question 78

If $(-3,-5)$ is on the terminal side of an angle $\theta$ in standard position, find $\sin \theta, \cos \theta$ , and $\tan \theta$ .

Question 79

If $\cos \theta=-\frac{7}{25}$ and $\theta$ is in quadrant III find the values of the other trigonometric functions of $\theta$ .

Question 80

Find the exact values of each part labeled with a letter.