5.1E: Exercises
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Verbal
1) Draw an angle in standard position. Label the vertex, initial side, and terminal side.
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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.
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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.
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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∘
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8) −80∘
9) 135∘
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10) −150∘
11) 2π3
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12) 7π4
13) 5π6
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14) π2
15) −π10
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16) 415∘
17) −120∘
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240∘
18) −315∘
19)22π3
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4π3
20) −π6
21) −4π3
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2π3
For the exercises 22-23, refer to Figure below. Round to two decimal places.
22) Find the arc length.
23) Find the area of the sector.
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27π2≈11.00 in2
For exercises 24-25, refer to Figure below. Round to two decimal places.
24) Find the arc length.
25) Find the area of the sector.
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81π20≈12.72 cm2
Algebraic
For exercises 26-32, convert angles in radians to degrees.
26) 3π4 radians
27) π9 radians
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20∘
28) −5π4 radians
29) π3 radians
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60∘
30) −7π3 radians
31) −5π12 radians
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−75∘
32) 11π6 radians
For exercises 33-39, convert angles in degrees to radians.
33) 90∘
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π2 radians
34) 100∘
35) −540∘
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−3π radians
36) −120∘
37) 180∘
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π radians
38) −315∘
39) 150∘
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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.
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5.02π3≈5.26 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∘.
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25π9≈8.73 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.
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21π10≈6.60 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.
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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.
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0.7697in2
For exercises 50-53, find the angle between 0∘ and 360∘ that is coterminal to the given angle.
50) −40∘
51) −110∘
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250∘
52) 700∘
53) 1400∘
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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
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4π3
56) 13π6
57) 44π9
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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?
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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?
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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.
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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.
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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?
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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.
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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.
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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?
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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?