Stewart Precalc 6e Section 6.1: Angle Measure

1 Concepts · Level 1

(a) The radian measure of an angle \(\theta\) is the length of the _____ that subtends the angle in a circle of radius _____.

Answer for 1(a)

(b) To convert degrees to radians, we multiply by _____.

Answer for 1(b)

(c) To convert radians to degrees, we multiply by _____.

Answer for 1(c)

Enter your answer directly below each part above.

2 Concepts · Level 1

A central angle \(\theta\) is drawn in a circle of radius \(r\).

(a) The length of the arc subtended by \(\theta\) is \(s =\) _____.

Answer for 2(a)

(b) The area of the sector with central angle \(\theta\) is \(A =\) _____.

Answer for 2(b)

Enter your answer directly below each part above.

3 Skills - Radian Measure · Level 1

Find the radian measure of the angle with the given degree measure: \(72^{\circ}\).

4 Skills - Radian Measure · Level 1

Find the radian measure of the angle with the given degree measure: \(-45^{\circ}\).

5 Skills - Radian Measure · Level 1

Find the radian measure of the angle with the given degree measure: \(-60^{\circ}\).

6 Skills - Radian Measure · Level 1

Find the radian measure of the angle with the given degree measure: \(-75^{\circ}\).

7 Skills - Radian Measure · Level 1

Find the radian measure of the angle with the given degree measure: \(-300^{\circ}\).

8 Skills - Radian Measure · Level 1

Find the radian measure of the angle with the given degree measure: \(1080^{\circ}\).

9 Skills - Radian Measure · Level 1

Find the radian measure of the angle with the given degree measure: \(3960^{\circ}\).

10 Skills - Radian Measure · Level 1

Find the radian measure of the angle with the given degree measure: \(96^{\circ}\).

11 Skills - Radian Measure · Level 1

Find the radian measure of the angle with the given degree measure: \(15^{\circ}\).

12 Skills - Radian Measure · Level 1

Find the radian measure of the angle with the given degree measure: \(7.5^{\circ}\).

13 Skills - Radian Measure · Level 1

Find the radian measure of the angle with the given degree measure: \(202.5^{\circ}\).

14 Skills - Degree Measure · Level 1

Find the degree measure of the angle with the given radian measure: \(\dfrac{7 \pi}{6}\).

15 Skills - Degree Measure · Level 1

Find the degree measure of the angle with the given radian measure: \(\dfrac{11 \pi}{3}\).

16 Skills - Degree Measure · Level 1

Find the degree measure of the angle with the given radian measure: \(-\dfrac{5 \pi}{4}\).

17 Skills - Degree Measure · Level 1

Find the degree measure of the angle with the given radian measure: \(-\dfrac{3 \pi}{2}\).

18 Skills - Degree Measure · Level 1

Find the degree measure of the angle with the given radian measure: \(3\).

19 Skills - Degree Measure · Level 1

Find the degree measure of the angle with the given radian measure: \(-2\).

20 Skills - Degree Measure · Level 1

Find the degree measure of the angle with the given radian measure: \(-1.2\).

21 Skills - Degree Measure · Level 1

Find the degree measure of the angle with the given radian measure: \(3.4\).

22 Skills - Degree Measure · Level 1

Find the degree measure of the angle with the given radian measure: \(\dfrac{\pi}{10}\).

23 Skills - Degree Measure · Level 1

Find the degree measure of the angle with the given radian measure: \(-\dfrac{2 \pi}{15}\).

24 Skills - Degree Measure · Level 1

Find the degree measure of the angle with the given radian measure: \(-\dfrac{13 \pi}{12}\).

25 Skills - Coterminal Angles · Level 2

The measure of an angle in standard position is \(50^{\circ}\). Find two positive angles and two negative angles that are coterminal with the given angle.

26 Skills - Coterminal Angles · Level 2

The measure of an angle in standard position is \(\dfrac{11 \pi}{6}\). Find two positive angles and two negative angles that are coterminal with the given angle.

27 Skills - Coterminal Angles · Level 2

The measure of an angle in standard position is \(-\dfrac{\pi}{4}\). Find two positive angles and two negative angles that are coterminal with the given angle.

28 Skills - Coterminal Angles · Level 2

The measure of an angle in standard position is \(-45^{\circ}\). Find two positive angles and two negative angles that are coterminal with the given angle.

29 Skills - Coterminal Determination · Level 2

Determine whether the angles \(70^{\circ}\) and \(430^{\circ}\) are coterminal.

30 Skills - Coterminal Determination · Level 2

Determine whether the angles \(-30^{\circ}\) and \(330^{\circ}\) are coterminal.

31 Skills - Coterminal Determination · Level 2

Determine whether the angles \(\dfrac{5 \pi}{6}\) and \(\dfrac{17 \pi}{6}\) are coterminal.

32 Skills - Coterminal Determination · Level 2

Determine whether the angles \(\dfrac{32 \pi}{3}\) and \(\dfrac{11 \pi}{3}\) are coterminal.

33 Skills - Coterminal Determination · Level 2

Determine whether the angles \(155^{\circ}\) and \(875^{\circ}\) are coterminal.

34 Skills - Coterminal Determination · Level 2

Determine whether the angles \(50^{\circ}\) and \(340^{\circ}\) are coterminal.

35 Skills - Coterminal in Range · Level 2

Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with \(733^{\circ}\).

36 Skills - Coterminal in Range · Level 2

Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with \(361^{\circ}\).

37 Skills - Coterminal in Range · Level 2

Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with \(1110^{\circ}\).

38 Skills - Coterminal in Range · Level 2

Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with \(-100^{\circ}\).

39 Skills - Coterminal in Range · Level 2

Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with \(-800^{\circ}\).

40 Skills - Coterminal in Range · Level 2

Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with \(1270^{\circ}\).

41 Skills - Coterminal in Range · Level 2

Find an angle between \(0\) and \(2 \pi\) that is coterminal with \(\dfrac{17 \pi}{6}\).

42 Skills - Coterminal in Range · Level 2

Find an angle between \(0\) and \(2 \pi\) that is coterminal with \(-\dfrac{7 \pi}{3}\).

43 Skills - Coterminal in Range · Level 2

Find an angle between \(0\) and \(2 \pi\) that is coterminal with \(10\).

44 Skills - Coterminal in Range · Level 2

Find an angle between \(0\) and \(2 \pi\) that is coterminal with \(\dfrac{17 \pi}{4}\).

45 Skills - Arc Length from Figure · Level 2

Find the length of the arc \(s\) in the figure.

46 Skills - Central Angle from Figure · Level 2

Find the angle \(\theta\) in the figure.

47 Skills - Radius from Figure · Level 2

Find the radius \(r\) of the circle in the figure.

48 Skills - Arc Length · Level 2

Find the length of an arc that subtends a central angle of \(45^{\circ}\) in a circle of radius 10 m.

49 Skills - Arc Length · Level 2

Find the length of an arc that subtends a central angle of 2 rad in a circle of radius 2 mi.

50 Skills - Central Angle from Arc · Level 2

A central angle \(\theta\) in a circle of radius 5 m is subtended by an arc of length 6 m. Find the measure of \(\theta\) in degrees and in radians.

51 Skills - Central Angle from Arc · Level 2

An arc of length 100 m subtends a central angle \(\theta\) in a circle of radius 50 m. Find the measure of \(\theta\) in degrees and in radians.

52 Skills - Radius from Arc · Level 2

A circular arc of length 3 ft subtends a central angle of \(25^{\circ}\). Find the radius of the circle.

53 Skills - Radius from Arc · Level 2

Find the radius of the circle if an arc of length 6 m on the circle subtends a central angle of \(\dfrac{\pi}{6}\) rad.

54 Skills - Radius from Arc · Level 2

Find the radius of the circle if an arc of length 4 ft on the circle subtends a central angle of \(135^{\circ}\).

55 Skills - Sector Area from Figure · Level 2

Find the area of the sector shown in each figure.

56 Skills - Radius from Sector Area · Level 2

Find the radius of each circle if the area of the sector is 12.

57 Skills - Sector Area · Level 2

Find the area of a sector with central angle 1 rad in a circle of radius 10 m.

58 Skills - Sector Area · Level 2

A sector of a circle has a central angle of \(60^{\circ}\). Find the area of the sector if the radius of the circle is 3 mi.

59 Skills - Radius from Sector Area · Level 2

The area of a sector of a circle with a central angle of 2 rad is 16 m². Find the radius of the circle.

60 Skills - Central Angle from Sector · Level 2

A sector of a circle of radius 24 mi has an area of 288 mi². Find the central angle of the sector.

61 Skills - Sector Area · Level 3

The area of a circle is 72 cm². Find the area of a sector of this circle that subtends a central angle of \(\dfrac{\pi}{6}\) rad.

62 Skills - Sector Area (Tangent Circles) · Level 4

Three circles with radii 1, 2, and 3 ft are externally tangent to one another, as shown in the figure. Find the area of the sector of the circle of radius 1 that is cut off by the line segments joining the center of that circle to the centers of the other two circles.

63 Applications - Travel Distance · Level 3

Travel Distance. A car's wheels are 28 in. in diameter. How far (in miles) will the car travel if its wheels revolve 10,000 times without slipping?

64 Applications - Wheel Revolutions · Level 3

Wheel Revolutions. How many revolutions will a car wheel of diameter 30 in. make as the car travels a distance of one mile?

65 Applications - Latitudes · Level 3

Latitudes. Pittsburgh, Pennsylvania, and Miami, Florida, lie approximately on the same meridian. Pittsburgh has a latitude of \(40.5^{\circ}\) N, and Miami has a latitude of \(25.5^{\circ}\) N. Find the distance between these two cities. (The radius of the earth is 3960 mi.)

66 Applications - Latitudes · Level 3

Latitudes. Memphis, Tennessee, and New Orleans, Louisiana, lie approximately on the same meridian. Memphis has a latitude of \(35^{\circ}\) N, and New Orleans has a latitude of \(30^{\circ}\) N. Find the distance between these two cities. (The radius of the earth is 3960 mi.)

67 Applications - Orbit · Level 3

Orbit of the Earth. Find the distance that the earth travels in one day in its path around the sun. Assume that a year has 365 days and that the path of the earth around the sun is a circle of radius 93 million miles. (The path of the earth around the sun is actually an ellipse with the sun at one focus. This ellipse, however, has very small eccentricity, so it is nearly circular.)

68 Applications - Eratosthenes · Level 4

Circumference of the Earth. The Greek mathematician Eratosthenes (ca. 276–195 B.C.) measured the circumference of the earth from the following observations. He noticed that on a certain day the sun shone directly down a deep well in Syene (modern Aswan). At the same time in Alexandria, 500 miles north (on the same meridian), the rays of the sun shone at an angle of \(7.2^{\circ}\) to the zenith. Use this information and the figure to find the radius and circumference of the earth.

69 Applications - Nautical Miles · Level 3

Nautical Miles. Find the distance along an arc on the surface of the earth that subtends a central angle of 1 minute (1 minute \(= \dfrac{1}{60}\) degree). This distance is called a nautical mile. (The radius of the earth is 3960 mi.)

70 Applications - Irrigation · Level 3

Irrigation. An irrigation system uses a straight sprinkler pipe 300 ft long that pivots around a central point as shown. Due to an obstacle the pipe is allowed to pivot through \(280^{\circ}\) only. Find the area irrigated by this system.

71 Applications - Windshield Wipers · Level 3

Windshield Wipers. The top and bottom ends of a windshield wiper blade are 34 in. and 14 in., respectively, from the pivot point. While in operation, the wiper sweeps through \(135^{\circ}\). Find the area swept by the blade.

72 Applications - Tethered Cow · Level 4

The Tethered Cow. A cow is tethered by a 100-ft rope to the inside corner of an L-shaped building, as shown in the figure. Find the area that the cow can graze.

73 Applications - Fan · Level 3

Fan. A ceiling fan with 16-in. blades rotates at 45 rpm.

(a) Find the angular speed of the fan in rad/min.

Answer for 73(a)

(b) Find the linear speed of the tips of the blades in in./min.

Answer for 73(b)

Enter your answer directly below each part above.

74 Applications - Radial Saw · Level 3

Radial Saw. A radial saw has a blade with a 6-in. radius. Suppose that the blade spins at 1000 rpm.

(a) Find the angular speed of the blade in rad/min.

Answer for 74(a)

(b) Find the linear speed of the sawteeth in ft/s.

Answer for 74(b)

Enter your answer directly below each part above.

75 Applications - Winch · Level 3

Winch. A winch of radius 2 ft is used to lift heavy loads. If the winch makes 8 revolutions every 15 s, find the speed at which the load is rising.

76 Applications - Speed of a Car · Level 3

Speed of a Car. The wheels of a car have radius 11 in. and are rotating at 600 rpm. Find the speed of the car in mi/h.

77 Applications - Equator Speed · Level 3

Speed at the Equator. The earth rotates about its axis once every 23 h 56 min 4 s, and the radius of the earth is 3960 mi. Find the linear speed of a point on the equator in mi/h.

78 Applications - Truck Wheels · Level 3

Truck Wheels. A truck with 48-in.-diameter wheels is traveling at 50 mi/h.

(a) Find the angular speed of the wheels in rad/min.

Answer for 78(a)

(b) How many revolutions per minute do the wheels make?

Answer for 78(b)

Enter your answer directly below each part above.

79 Applications - Speed of Current · Level 3

Speed of a Current. To measure the speed of a current, scientists place a paddle wheel in the stream and observe the rate at which it rotates. If the paddle wheel has radius 0.20 m and rotates at 100 rpm, find the speed of the current in m/s.

80 Applications - Bicycle Wheel · Level 4

Bicycle Wheel. The sprockets and chain of a bicycle are shown in the figure. The pedal sprocket has a radius of 4 in., the wheel sprocket a radius of 2 in., and the wheel a radius of 13 in. The cyclist pedals at 40 rpm.

(a) Find the angular speed of the wheel sprocket.

Answer for 80(a)

(b) Find the speed of the bicycle. (Assume that the wheel turns at the same rate as the wheel sprocket.)

Answer for 80(b)

Enter your answer directly below each part above.

81 Applications - Conical Cup · Level 4

Conical Cup. A conical cup is made from a circular piece of paper with radius 6 cm by cutting out a sector and joining the edges as shown on the next page. Suppose \(\theta = \dfrac{5 \pi}{3}\).

(a) Find the circumference \(C\) of the opening of the cup.

Answer for 81(a)

(b) Find the radius \(r\) of the opening of the cup. [Hint: Use \(C = 2 \pi r\).]

Answer for 81(b)

(c) Find the height \(h\) of the cup. [Hint: Use the Pythagorean Theorem.]

Answer for 81(c)

Enter your answer directly below each part above.

82 Exercise - Conical Cup · Level 3

Conical Cup. In this exercise we find the volume of the conical cup in Exercise 87 for any angle \(\theta\).

(a) Follow the steps in Exercise 87 to show that the volume of the cup as a function of \(\theta\) is \( V(\theta) = \dfrac{9}{\pi^2} \theta^2 \sqrt{4 \pi^2 - \theta^2}, 0 < \theta < 2 \pi \)

Answer for 82(a)

(b) Graph the function \(V\).

Answer for 82(b)

(c) For what angle \(\theta\) is the volume of the cup a maximum?

Answer for 82(c)

Enter your answer directly below each part above.

83 Exercise - Discovery/Discussion/Writing · Level 2

Different Ways of Measuring Angles. The custom of measuring angles using degrees, with \(360^{\circ}\) in a circle, dates back to the ancient Babylonians, who used a number system based on groups of 60. Another system of measuring angles divides the circle into 400 units, called grads. In this system a right angle is 100 grad, so this fits in with our base 10 number system. Write a short essay comparing the advantages and disadvantages of these two systems and the radian system of measuring angles. Which system do you prefer? Why?

84 Exercise - Clocks and Angles · Level 2

Clocks and Angles. In one hour, the minute hand on a clock moves through a complete circle, and the hour hand moves through \(\dfrac{1}{12}\) of a circle. Through how many radians do the minute and the hour hand move between 1:00 P.M. and 6:45 P.M. (on the same day)?

85 Example - Converting Between Radians and Degrees · Level 1

*(a)* Express \(60^{\circ}\) in radians. *(b)* Express \(\dfrac{\pi}{6}\) rad in degrees.

86 Example - Coterminal Angles · Level 2

*(a)* Find angles that are coterminal with the angle \(\theta = 30^{\circ}\) in standard position. *(b)* Find angles that are coterminal with the angle \(\theta = \dfrac{\pi}{3}\) in standard position.

87 Example - Coterminal Angles · Level 2

Find an angle with measure between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the angle of measure \(1290^{\circ}\) in standard position.

88 Example - Arc Length and Angle Measure · Level 2

*(a)* Find the length of an arc of a circle with radius 10 m that subtends a central angle of \(30^{\circ}\). *(b)* A central angle \(\theta\) in a circle of radius 4 m is subtended by an arc of length 6 m. Find the measure of \(\theta\) in radians.

89 Example - Area of a Sector · Level 2

Find the area of a sector of a circle with central angle \(60^{\circ}\) if the radius of the circle is 3 m.

90 Example - Linear and Angular Speed · Level 2

A boy rotates a stone in a 3-ft-long sling at the rate of 15 revolutions every 10 seconds. Find the angular and linear velocities of the stone.

91 Example - Linear Speed from Angular Speed · Level 3

A woman is riding a bicycle whose wheels are 26 inches in diameter. If the wheels rotate at 125 revolutions per minute (rpm), find the speed at which she is traveling, in mi/h.

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