Chapter 5 Introduction to Trigonometric Functions

Transcription

1 Chapter 5 Introduction to Trigonometric Functions 5.1 Angles Section Exercises Verbal 1. Draw an angle in standard position. Label the vertex, initial side, and terminal side. 2. Explain why there are an infinite number of angles that are coterminal to a certain angle. Coterminal angles can be found by adding or subtracting any multiple of 360 or 2. These additions and subtractions are infinite. 3. State what a positive or negative angle signifies, and explain how to draw each. 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. There are 360 degrees and 2 radians in a full circle. Radian measure is directly 1 related to the circumference of a circle relative to its radius. One degree is 360 of a circle, one radian is formed from the angle that creates an arc equal to the length of the radius. 5. Explain the differences between linear speed and angular speed when describing motion along a circular path. 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 the following exercises, draw an angle in standard position with the given measure

2

3

4

5

6

7 = 3 22 " 6 # % & = 3 ' 3 (

8 = =

9 2 3 For the following exercises, refer to the figure below. Round to two decimal places. 22. Find the arc length = = in Find the area of the sector Area of sector = θr = 3 =

10 in 2 2 For the following exercises, refer to the figure below. Round to two decimal places. 24. Find the arc length = 4.5 = cm Find the area of the sector. Area of sector cm # $ = θr = 4.5 = = & ' ( 2 ) Algebraic For the following exercises, convert angles in radians to degrees radians

11 27. 9 radians radians radians radians radians radians For the following exercises, convert angles in degrees to radians

12 2 radians radians radians radians radians radians radians For the following exercises, use the 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 4. radians. = inches

13 41. Find the length of the arc of a circle of radius 5.02 miles subtended by the central angle of 3. = miles 42. Find the length of the arc of a circle of diameter 14 meters subtended by the central 5 angle of 6. d = 14 = 2r r = 7 5 = miles 43. Find the length of the arc of a circle of radius 10 centimeters subtended by the central 50. angle of 5 50 = = centimeters 44. Find the length of the arc of a circle of radius 5 inches subtended by the central angle of = = meters 45. Find the length of the arc of a circle of diameter 12 meters subtended by the central angle is 63.

14 d = 2r = 12 r = = = meters For the following exercises, 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 and a radius 6 cm. 45 = Area of sector cm A sector of a circle has a central angle of and a radius of 20 cm. 30 = Area of sector cm = θr = = 2 2 = θr = = 48. A sector of a circle with diameter 10 feet and an angle of 2 radians. d = 2r = 10 r = Area of sector = θr = 5 = ft A sector of a circle with radius of 0.7 inches and an angle of radians Area of sector = θr = 0.7 = in 2 For the following exercises, find the angle between and that is coterminal to the given angle

15 For the following exercises, find the angle between 0 and the given angle = = = ( ) ( 2 ) " 2 # = % & = 9 9 ' 9 ( in radians that is coterminal to

16 8 9 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? d = 2r = 32 r = 16 in C = 2 r = 2 16 = 32 in 60 mi 5280 ft 12 in 1 h = in/min 1 h 1 mi 1 ft 60 min in 1 rev = rev/min 1 min 32 in rev 2 rad = 3960 rad/min 1 min 1 rev 3960 rad/min RPM 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? d = 2r = 24 r = 12 in C = 2 r = 2 12 = 24 in = 2 ft 15 mi 5280 ft 1 h = 1320 ft/min 1 h 1 mi 60 min 1320 ft 1 rev rev/min 1 min 2 ft rev 2 rad = 1320 rad/min 1 min 1 rev 1320 rad/min 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/sec? 8 in 15 /s in/s rot 60 s = 2.5 rot/min 1 s min 15 = rad/s 1 s in/s, 2.5 RPM, 12 rad/s

17 61. A wheel of radius 14 inches is rotating 0.5 rad/s. What is the linear speed v, the angular speed in RPM, and the angular speed in deg/s? 0.5 rad 14 in = 7 in/s 1 s 0.5 rad deg/s 1 s 7 in/s, v RPM, 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. 200 rot 2 60 mm 1 min 75, mm/min = m/sec 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 rot 2 60 mm 1 min 1,809, mm/min = m/s 64. A person is standing on the equator of Earth (radius 3960 miles). What are his linear and angular speeds? 2 rad = rad/h 24 h 12 rad 3960 mi h rad/h. Angular speed: 12 Linear speed: miles/h 65. Find the distance along an arc on the surface of Earth that subtends a central angle of 1 1 minute = degree 5 minutes ( 60 ). The radius of Earth is 3960 miles. # 5 $ = 3960 & ' ( 60 ) miles 66. Find the distance along an arc on the surface of Earth that subtends a central angle of 1 1 minute = degree 7 minutes ( 60 ). The radius of Earth is 3960miles. # 7 $ = 3960 & ' ( 60 ) 180

18 8.06 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? 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 degree north. Assume the radius of the earth is 3960 miles. Find the distance between the two cities = θ = = = miles 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. r = cos = ( ) C = 2 r = = mi mi/h 24 h 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. r = cos = ( ) C = 2 r = = mi mi/h 24 h 268 miles per hour 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.

19 1 θ = 2 = = 239, mi mi/h 24 h 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 travelling down the road. 180 rot 60 min 1 ft 1 mi 2 14 in mi/h 1 min 1 h 12 in 5280 ft miles per hour 73. A car travels 3 miles. Its tires make 2640 revolutions. What is the radius of a tire in inches? 3 mi 5280 ft 12 in = 72 in/rev 2640 rev 1 mi 1 ft 72=2 r 72 r = 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? 5280 ft 12 in 4 mi = 253,440 in 1 mi 1 ft C = 2 r = 2 12 = 24 in 253,440 in in 3361 revolutions This file is copyright , Rice University. All Rights Reserved.