Trigonometry Section 3.1
Convert each degree measure to radians.
1. 23 .
0143

2. 30

5. 85 .
04

6. 120

3.
7.
39

174

5 0

4.
8.
42

3
270

0

9. 315

10. 390

11. 450

12. 600

Convert each radian measure to degrees. Write answers to the nearest minute or thousandths of a degree.
13.

3 .
47189 14.
4

15
15. 0 .
3417 16.
7

10
4

7

11

17. 18. 1 .
74 19. 20.
3 4 6
5

8

15

21. 22. 23. 24. 5
2 3 4
Find the EXACT value of each of the following WITHOUT using a calculator.
2

5
 
25. sin 26. cos 27. tan 28.
3 6 4
29. sec
3

4
30. csc
7

6
31. cos

4

3
32.
33. csc

11

4
34. tan
6

35. csc

2
36.
23

13

37. cos 3

38. cot
 6
39. sin
4
40. cot
5

3 sin sec
11

6 sec


13

3

Trigonometry Section 3.2
Find the length of the arc intercepted by a central angle
 in a circle of radius r .
1. r

12 .
3 cm ,
 
2

3 radians 2. r

0 .
892 cm ,
 
11

10 radians
3. r

253 m ,
 
2

5 radians 4. r

120 mm ,
 

9 radians
5. r

4 .
82 m ,
 
60

6. r

71 .
9 cm ,
 
135

Find the measure (in radians) of a central angle that intercepts an arc of given length in a circle with given radius.
7. s

5 in , r

2 in 8. s

6 m , r

4 m 9. s

30 cm , r

5 cm
Find the radius of a circle in which a central angle (in radians) is given in radians and intercepts an arc of given length.
10.
 
2 radians , s

3 ft 11.
 
3

4 radians , s

6
 cm
12.
 
 radians , s

4 in
5
Find the distance in kilometers between each pair of cities, assuming they lie on the same northsouth line.
13. Panama City, Panama 9
Pittsburgh, Pennsylvania

N
40

N
15.
14. New York City, New York
Lima, Peru 12

S
41

N
Assuming that Earth is a sphere of radius 4000 miles, what is the difference in the latitudes of Syracuse, New York and Annapolis, Maryland, where Syracuse is about 450 miles due north of Annapolis?
Work the applied problems.
16. a) How many inches will the weight in the figure rise if the pulley is rotated through an angle of 71

5 0

? b) Through what angle, to the nearest minute, must the pulley be rotated to raise the weight 6 in ?
9.27 in

17. The rotation of the smaller wheel in the figure causes the larger wheel to rotate. Through how many degrees will the larger wheel rotate if the smaller one rotates through 60 .
0

?
5.23cm 8.16cm
Find the area of a sector of a circle having radius
18. r

29 .
2 m ,
 
5

6 radians r and central angle
19. r


.
59 .
8 km ,
 
20. r

52 cm ,
 
3

10 radians 21. r

25 mm ,
 
2

3 radians

15 radians
22. r

12 .
7 cm ,
 
81

23. r

18 .
3 m ,
 
125

Find the measure (in radians) of a central angle of a sector of given area in a circle with given radius.
24. A

16 in
2
, r

3 .
0 in 25. A

25 in
2
, r

10 in
Find the radius of a circle in which a central angle is given in radians and determines a sector of given area.
26.
 

4 radians , A

36 ft 2
27.
 

6 radians , A

64 m 2
Work the applied problem.
28. The figure shows Medicine Wheel, a Native American structure in northern Wyoming.
This circular structure is perhaps 200 years old. There are 32 spokes in the wheel, all equally spaced. a) Find the measure of each central angle in degrees and radians. b) If the radius of the wheel is 76 ft , find the circumference. c) Find the length of each arc intercepted by consecutive pairs of spokes. d) Find the area of each sector formed by consecutive spokes.
29. A sprinkler on a golf green is set to spray water over a distance of 15 meters and to rotate through an angle of 140

. Find the area of the region.

Trigonometry Section 3.3
Find the EXACT circular function value for each of the following WITHOUT using a calculator.
5

3

4

7

1. sin
6
2. cos
4
3. tan
3
4. cot

4
5. sec 2

6. csc
17

3
7. cos

5

2
8. sec
8

3
5

5

7

9. tan
2
10. sin
3
11. cot
 
12. csc
 6
13. cos
7

6
14. sec

3

2
15. csc


3
 
16. sin
3

2
17.
24. cot
11

6 tan 0 .
9047
18. tan
25.
2

19. sec
13

4
26.
20. cos

Use a calculator to find an approximation for each circular function value.
21. sin 7 .
5835 22. cot 0 .
6632 23. cos 3 .
8426 csc 1 .
3875 sec 6 .
6701
27. sin


2 .
2864

28. tan 6 .
4752 29. cot


7 .
4526

9

4
Find the value of s in the interval

 0 ,

2

 that makes each statement true.
30. sin s

0 .
99184065 31. cot s

0 .
29949853
32. cos s

0 .
78269876 33. csc s

1 .
0219553
34. tan s

0 .
21264138 35. sec s

1 .
0219553
Find the EXACT value of s in the given interval that has the given circular function value. Do
NOT use a calculator.
36. sin s

1
2
,



2
,


 37. cos s


1
,
2



,
3

2

 38. tan s
 
1 ,


3

2
, 2



For each value of s , use a calculator to find sin s and cos s , then use the results to decide in which quadrant an angle of s radians lies.
39. s

49 40. s

65

Trigonometry Section 3.4
Use the formulas to find the value of variables.
1. Find
 if
 

4 radians / sec, t

5 min
2. Find
 if
 
2

5
,

10sec
3. Find t if
 
3

8 radians ,
 

24 radians / min
4. Find
 if
 
0 .
90674 radians / min, t

11 .
876 min
2

5. Find v if r

12 m ,
  radians / sec
3
6. Find
 if v

9 m / sec, r

5 m
14.
7. Find
 if v

107 .
692 m / sec, r

58 .
7413 m
8.
9.
Find s if r
Find t if s


6 cm ,

12

5 m , r


3
 radians / sec, t
3
2 m ,
 
2

5

9 sec radians / sec
10.
12.
Find
 if s

3
 km , r

2 km , t

4 sec
4
Work the following problems.
11. Find
 for the minute hand of a clock.
Find v for the tip of the minute hand of a clock, if the hand is 7 cm long.
13. Find
 for a line from the center to the edge of a phonograph record revolving
1
33
3 times per minute.
Find v for a point on the tread of a tire 18 cm , rotating 35 times per minute.

15.
16.
17.
The Earth travels about the Sun in an orbit that is almost circular. Assume that the orbit is a circle, with a radius 93 , 000 , 000 miles b)
. a) Assume that a year is 365 days, and find

, the angle formed by the Earth’s movement in one day.
Give the angular velocity in radians per hour. c) Find the linear velocity of the Earth in miles per hour.
A circular power saw has a 7
1
4 inch diameter blade that rotates at 5000 revolutions per minute. a) Find the angular velocity of the saw blade in radians per minute. b) Find the linear velocity (in feet per minute) of one of the 24 cutting teeth as they contact the wood being cut.
A car is moving at a rate of 65 miles per hour and the diameter of its wheels is 2 feet. a) Find the number of revolutions per minute the wheels are rotating. b) Find the angular velocity of the wheels in radians per minute.

TRIGONOMETRY NAME: ____________________
PRACTICE TEST: Radian Measure and the Circular Functions
Chapter 3
In order to receive full credit, you must show your work!!
Convert each of the following degree measures to radians. Leave answers as multiples of

.
1. 45

2. 210

3. 1020

Convert each of the following radian measures to degrees.
11

8

4. 5.
6 3 5
Find the EXACT value of each of the following WITHOUT using a calculator.
5
 
7. cos 8. sin 9. tan
4 3
10. csc
3

2
11. cot
4

3
6.
12.
21

5

6 sec
2

3
13. tan
7

14. sin
11

15. cos

4 6
Use a calculator to find an approximation for the circular function values. Be sure your calculator is set in radian mode.
16. sin 1 .
0472 17.
Find the value of s in the interval

 0 ,

2

 sec 0 .
4864 18. cot 3
that makes each of the following true.
.
8426
19. cos s

0 .
92500448 20. csc s

1 .
2361343 21. tan s

4 .
0112357
Find the EXACT value of s in the given interval that has the given circular function value. Do
NOT use a calculator.
22. cos s


2
2
,



2
,


 23. tan s

3
3
,



,
3

2


24. sin s


2
3
,


3

2
, 2




32.
Solve each problem.
25. A circle has a radius of 8.973cm. Find the length of an arc on this circle intercepted by a central angle of 49 .
06

.
26.
7

A central angle of radians forms a sector of a circle. Find the area of the sector if
4 the radius of the circle is 28.69in.
27. Find the measure of the central angle
  radians

AND the area of the sector for a circle with a radius of 2m and an arc length of 1.5m.
28. Find the distance in kilometers between Farmersville, California,
British Columbia, 49

N
36

N and Penticton,
, assuming they lie on the same north-south line. (Assume that the radius of the Earth is 6400km)
29. Find s if r

11 .
46 cm ,
 
4 .
283 radians / sec, and t

5 .
813 sec .
30. Find

if v

18 ft / sec and r

3 ft .
31. Find r if s

216

5 yd ,
 
2

5 radians / sec and t

12 sec .
Find v for the tip of an airplane propeller 3m long, rotating 500 times per min.

17.
19.
Solutions

1.
4
5. 480

8. sin


3
3 2
2.
6.
11.
14. cot
4


3
3 3 sin
11

6
 
1
2 s

0.3897489421
7

6
756

9.
12.
15.
21. s

1.326476844
3.
7. tan
5

6
 
3
3 sec
2

3
 
2 cos
22.
  
1
17

3
4. cos
5

4
 
2
2
16.
10.
13. sin1.0472

18.
20. s

3

4 s

0.9424039526
23.
330
 csc
3

2
 
1 tan
7

4
 
1
0.8660266282
s

7

6
24. s

5

3
25. s

7.683
cm 26. A

2262.22

2263 in 2
27.
 
0.75
,

1.5
m
2
28. s
  km
29. s

285.32

285.3
cm 30.
 
6 rads sec
31. r

9 yds 32. v

4712

4700 m min