Loading [MathJax]/jax/element/mml/optable/Latin1Supplement.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

5.1E: Exercises

( \newcommand{\kernel}{\mathrm{null}\,}\)

Verbal

1) Draw an angle in standard position. Label the vertex, initial side, and terminal side.

Answer

Ex 5.1.1.png

2) Explain why there are an infinite number of angles that are coterminal to a certain angle.

3) State what a positive or negative angle signifies, and explain how to draw each.

Answer

Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.

4) How does radian measure of an angle compare to the degree measure? Include an explanation of 1 radian in your paragraph.

5) Explain the differences between linear speed and angular speed when describing motion along a circular path.

Answer

Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.

Graphical

For exercises 6-21, draw an angle in standard position with the given measure.

6) 30

7) 300

Answer

CNX_Precalc_Figure_05_01_203a.jpg

8) 80

9) 135

Answer

CNX_Precalc_Figure_05_01_205.jpg

10) 150

11) 2π3

Answer

CNX_Precalc_Figure_05_01_207.jpg

12) 7π4

13) 5π6

Answer

CNX_Precalc_Figure_05_01_209.jpg

14) π2

15) π10

Answer

CNX_Precalc_Figure_05_01_211.jpg

16) 415

17) 120

Answer

240

CNX_Precalc_Figure_05_01_213.jpg

18) 315

19)22π3

Answer

4π3

CNX_Precalc_Figure_05_01_215.jpg

20) π6

21) 4π3

Answer

2π3

CNX_Precalc_Figure_05_01_217.jpg

For the exercises 22-23, refer to Figure below. Round to two decimal places.

CNX_Precalc_Figure_05_01_218.jpg

22) Find the arc length.

23) Find the area of the sector.

Answer

27π211.00 in2

For exercises 24-25, refer to Figure below. Round to two decimal places.

CNX_Precalc_Figure_05_01_219.jpg

24) Find the arc length.

25) Find the area of the sector.

Answer

81π2012.72 cm2

Algebraic

For exercises 26-32, convert angles in radians to degrees.

26) 3π4 radians

27) π9 radians

Answer

20

28) 5π4 radians

29) π3 radians

Answer

60

30) 7π3 radians

31) 5π12 radians

Answer

75

32) 11π6 radians

For exercises 33-39, convert angles in degrees to radians.

33) 90

Answer

π2 radians

34) 100

35) 540

Answer

3π radians

36) 120

37) 180

Answer

π radians

38) 315

39) 150

Answer

5π6 radians

For exercises 40-45, use to given information to find the length of a circular arc. Round to two decimal places.

40) Find the length of the arc of a circle of radius 12 inches subtended by a central angle of π4 radians.

41) Find the length of the arc of a circle of radius 5.02 miles subtended by the central angle of π3.

Answer

5.02π35.26 miles

42) Find the length of the arc of a circle of diameter 14 meters subtended by the central angle of 5π6.

43) Find the length of the arc of a circle of radius 10 centimeters subtended by the central angle of 50.

Answer

25π98.73 centimeters

44) Find the length of the arc of a circle of radius 5 inches subtended by the central angle of 220circ.

45) Find the length of the arc of a circle of diameter 12 meters subtended by the central angle is 63circ.

Answer

21π106.60 meters

For exercises 46-49, use the given information to find the area of the sector. Round to four decimal places.

46) A sector of a circle has a central angle of 45 and a radius 6 cm.

47) A sector of a circle has a central angle of 30 and a radius of 20 cm.

Answer

104.7198cm2

48) A sector of a circle with diameter 10 feet and an angle of π2 radians.

49) A sector of a circle with radius of 0.7 inches and an angle of π radians.

Answer

0.7697in2

For exercises 50-53, find the angle between 0 and 360 that is coterminal to the given angle.

50) 40

51) 110

Answer

250

52) 700

53) 1400

Answer

320

For exercises 54-57, find the angle between 0 and 2π in radians that is coterminal to the given angle.

54) π9

55) 10π3

Answer

4π3

56) 13π6

57) 44π9

Answer

8π9

Real-World Applications

58) A truck with 32-inch diameter wheels is traveling at 60 mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make?

59) A bicycle with 24-inch diameter wheels is traveling at 15 mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make?

Answer

1320 rad 210.085 RPM

60) A wheel of radius 8 inches is rotating 15/s. What is the linear speed v, the angular speed in RPM, and the angular speed in rad/s?

61) A wheel of radius 14 inches is rotating 0.5rad/s. What is the linear speed v, the angular speed in RPM, and the angular speed in deg/s?

Answer

7 in./s, 4.77 RPM, 28.65 deg/s

62) A CD has diameter of 120 millimeters. When playing audio, the angular speed varies to keep the linear speed constant where the disc is being read. When reading along the outer edge of the disc, the angular speed is about 200 RPM (revolutions per minute). Find the linear speed.

63) When being burned in a writable CD-R drive, the angular speed of a CD is often much faster than when playing audio, but the angular speed still varies to keep the linear speed constant where the disc is being written. When writing along the outer edge of the disc, the angular speed of one drive is about 4800 RPM (revolutions per minute). Find the linear speed if the CD has diameter of 120 millimeters.

Answer

1,809,557.37 mm/min=30.16 m/s

64) A person is standing on the equator of Earth (radius 3960 miles). What are his linear and angular speeds?

65) Find the distance along an arc on the surface of Earth that subtends a central angle of 5 minutes (1 minute=160 degree). The radius of Earth is 3960 miles.

Answer

5.76 miles

66) Find the distance along an arc on the surface of Earth that subtends a central angle of 7 minutes (1 minute=160 degree). The radius of Earth is 3960 miles.

67) Consider a clock with an hour hand and minute hand. What is the measure of the angle the minute hand traces in 20 minutes?

Answer

120°

Extensions

68) Two cities have the same longitude. The latitude of city A is 9.00 degrees north and the latitude of city B is 30.00 degree north. Assume the radius of the earth is 3960 miles. Find the distance between the two cities.

69) A city is located at 40 degrees north latitude. Assume the radius of the earth is 3960 miles and the earth rotates once every 24 hours. Find the linear speed of a person who resides in this city.

Answer

794 miles per hour

70) A city is located at 75 degrees north latitude. Assume the radius of the earth is 3960 miles and the earth rotates once every 24 hours. Find the linear speed of a person who resides in this city.

71) Find the linear speed of the moon if the average distance between the earth and moon is 239,000 miles, assuming the orbit of the moon is circular and requires about 28 days. Express answer in miles per hour.

Answer

2,234 miles per hour

72) A bicycle has wheels 28 inches in diameter. A tachometer determines that the wheels are rotating at 180 RPM (revolutions per minute). Find the speed the bicycle is traveling down the road.

73) A car travels 3 miles. Its tires make 2640 revolutions. What is the radius of a tire in inches?

Answer

11.5 inches

74) A wheel on a tractor has a 24-inch diameter. How many revolutions does the wheel make if the tractor travels 4 miles?


5.1E: Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

  • Was this article helpful?

Support Center

How can we help?