Section 4.2: Radians, Arc Length, and the Area of a Sector
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1 CHAPTER 4 Trigonometric Functions Section 4.: Radians, Arc Length, and the Area of a Sector Measure of an Angle Formulas for Arc Length and Sector Area Measure of an Angle Degree Measure: 368
2 SECTION 4. Radians, Arc Length, and the Area of a Sector Radian Measure: Example: MATH 1330 Precalculus 369
3 CHAPTER 4 Trigonometric Functions Converting Between Degree Measure and Radian Measure: 370
4 SECTION 4. Radians, Arc Length, and the Area of a Sector Example: MATH 1330 Precalculus 371
5 CHAPTER 4 Trigonometric Functions Additional Example 1: 37
6 SECTION 4. Radians, Arc Length, and the Area of a Sector Additional Example : Part (a): Part (b): Part (c): MATH 1330 Precalculus 373
7 CHAPTER 4 Trigonometric Functions Additional Example 3: Part (a): Part (b): Part (c): 374
8 SECTION 4. Radians, Arc Length, and the Area of a Sector Additional Example 4: Formulas for Arc Length and Sector Area Arc Length: MATH 1330 Precalculus 375
9 CHAPTER 4 Trigonometric Functions Example: 376
10 SECTION 4. Radians, Arc Length, and the Area of a Sector Area of a Sector of a Circle: Example: MATH 1330 Precalculus 377
11 CHAPTER 4 Trigonometric Functions Example: Part (a): 378
12 SECTION 4. Radians, Arc Length, and the Area of a Sector Part (b): Part (c): Additional Example 1: MATH 1330 Precalculus 379
13 CHAPTER 4 Trigonometric Functions Additional Example : 380
14 Exercise Set 4.: Radians, Arc Length, and the Area of a Sector If we use central angles of a circle to analyze angle measure, a radian is an angle for which the arc of the circle has the same length as the radius, as illustrated in the figures below. Answer the following. 7. In the figure below, 1 radian. r r r r 1 radian, since the arc length is the same as the length of the radius. Note: Standard notation is to say 1; radians are implied when there is no angle measure. radians, since the arc length is twice the length of the radius. Note: Standard notation is to say. (a) Use this figure as a guide to sketch and estimate the number of radians in a complete revolution. (b) Give an exact number for the number of radians in a complete revolution. Justify your answer. Then round this answer to the nearest hundredth and compare it to the result from part (a). The number of radians can therefore be determined s by dividing the arc length s by the radius r, i.e.. r Find in the examples below cm 7 cm 8. Fill in the blanks: (a) 360 (b) 180 radians radians Convert the following degree measures to radians. First, give an exact result. Then round each answer to the nearest hundredth. 9. (a) 30 (b) 90 (c) (a) 45 (b) 60 (c) (a) 10 (b) 5 (c) ft 10 ft 1. (a) 10 (b) 70 (c) (a) 19 (b) 40 (c) (a) 10 (b) 53 (c) 88 Convert the following radian measures to degrees. 15. (a) 4 (b) 4 3 (c) r 0.3 m; s 7 cm 4. r 0.6 in; s 3.06 in 5. r 64 ft; s 3 ft 16. (a) (b) (a) (b) 11 1 (c) (c) r 60 in; s 1 ft 18. (a) 9 (b) 7 18 (c) MATH 1330 Precalculus 381
15 Exercise Set 4.: Radians, Arc Length, and the Area of a Sector Convert the following radian measures to degrees. Round answers to the nearest hundredth. 19. (a).5 (b) (a) 3.8 (b) 0.97 Answer the following. 1. If two angles of a triangle have radian measures and, find the radian measure of the third 1 5 angle.. If two angles of a triangle have radian measures 3 and, find the radian measure of the third 9 8 angle. In numbers 31-34, change to radians and then find the arc length using the formula s r. Compare results with those from exercises ; r 1 cm ; r 10 in ; r 4 ft ; r 1 cm Find the missing measure in each example below s 9 in To find the length of the arc of a circle, think of the arc length as simply a fraction of the circumference of the circle. If the central angle defining the arc is given in degrees, then the arc length can be found using the formula: s r s 3 m 80 Use the formula above to find the arc length s ; r 1 cm ; r 10 in 5. 5 ; r 4 ft ; r 1 cm If the central angle defining the arc is instead given in radians, then the arc length can be found using the formula: s r r Use the formula s r to find the arc length s: ; r 9 yd 6 3 ; r 6 cm 4 9. ; r ft ; r 30 in r 6 cm; 300 ; s? r 10 ft; ; s 4? s m; ; r 5? s in; ; r 3 6? s 7 ft; ; r 4? 4. s 0 cm; ; r 3? 43. s1 in; 5; r? 44. s 7 in; 3; r? 45. s 15 m; 70 ; r? 46. s 8 yd; 5 ; r? 38
16 Exercise Set 4.: Radians, Arc Length, and the Area of a Sector Answer the following. 47. Find the perimeter of a sector of a circle with central angle 6 and radius 8 cm. 48. Find the perimeter of a sector of a circle with central angle 7 and radius 3 ft. 4 To find the area of a sector of a circle, think of the sector as simply a fraction of the circle. If the central angle defining the sector is given in degrees, then the area of the sector can be found using the formula: A r 360 Use the formula above to find the area of the sector: ; r 1 cm ; r 10 in ; r 4 ft ; r 1 cm If the central angle defining the sector is instead given in radians, then the area of the sector can be found using the formula: A 1 r r 1 Use the formula A r to find the area of the sector: ; r 9 yd 6 3 ; r 6 cm ; r ft ; r 30 in 3 In numbers 57-60, change to radians and then find 1 the area of the sector using the formula A r. Compare results with those from exercises ; r 1 cm ; r 10 in ; r 4 ft ; r 1 cm Answer the following. 61. A sector of a circle has central angle 4 and area 49 cm. Find the radius of the circle A sector of a circle has central angle 5 and 6 5 area ft. Find the radius of the circle A sector of a circle has central angle 10 and 16 area in. Find the radius of the circle A sector of a circle has central angle 10 and 1 area m. Find the radius of the circle A sector of a circle has radius 6 ft and area 63 ft. Find the central angle of the sector (in radians). 66. A sector of a circle has radius 4 cm and area 7 8 cm. Find the central angle of the sector (in 49 radians). Answer the following. SHOW ALL WORK involved in obtaining each answer. Give exact answers unless otherwise indicated. 67. A CD has a radius of 6 cm. If the CD s rate of turn is 900 o /sec, find the following. (a) The angular speed in units of radians/sec. (b) The linear speed in units of cm/sec of a point on the outer edge of the CD. (c) The linear speed in units of cm/sec of a point halfway between the center of the CD and its outer edge. 68. Each blade of a fan has a radius of 11 inches. If the fan s rate of turn is 1440 o /sec, find the following. (a) The angular speed in units of radians/sec. (b) The linear speed in units of inches/sec of a point on the outer edge of the blade. (c) The linear speed in units of inches/sec of a point on the blade 7 inches from the center. MATH 1330 Precalculus 383
17 Exercise Set 4.: Radians, Arc Length, and the Area of a Sector 69. A bicycle has wheels measuring 6 inches in diameter. If the bicycle is traveling at a rate of 0 miles per hour, find the wheels rate of turn in revolutions per minute (rpm). Round the answer to the nearest hundredth. 70. A car has wheels measuring 16 inches in diameter. If the car is traveling at a rate of 55 miles per hour, find the wheels rate of turn in revolutions per minute (rpm). Round the answer to the nearest hundredth. 71. A car has wheels with a 10 inch radius. If each wheel s rate of turn is 4 revolutions per second, (a) Find the angular speed in units of radians/second. (b) How fast is the car moving in units of inches/sec? (c) How fast is the car moving in miles per hour? Round the answer to the nearest hundredth. 7. A bicycle has wheels with a 1 inch radius. If each wheel s rate of turn is revolutions per second, (a) Find the angular speed in units of radians/second. (b) How fast is the bicycle moving in units of inches/sec? (c) How fast is the bicycle moving in miles per hour? Round the answer to the nearest hundredth. 73. A clock has an hour hand, minute hand, and second hand that measure 4 inches, 5 inches, and 6 inches, respectively. Find the distance traveled by the tip of each hand in 0 minutes. 74. An outdoor clock has an hour hand, minute hand, and second hand that measure 1 inches, 14 inches, and 15 inches, respectively. Find the distance traveled by the tip of each hand in 45 minutes. 384
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