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4-2 Degrees and Radians 4-2 Degrees and Radians Write each degree measure in radians as a multiple of π and each radian measure in degrees. 10. 30° SOLUTION: To convert a degree measure to radians, multiply by ANSWER: 14. SOLUTION: To convert a radian measure to degrees, multiply by ANSWER: 120° Identify all angles that are coterminal with the given angle. Then find and draw one positive and one negative angle coterminal with the given angle. 20. 225° SOLUTION: All angles measuring are coterminal with a angle. Sample answer: Let n = 1 and −1. eSolutions Manual - Powered by Cognero Page 1 ANSWER: 225° + 360n°; Sample answer: 585°, −135° 24. SOLUTION: All angles measuring are coterminal with a angle. Sample answer: Let n = 1 and −1. ANSWER: Find the length of the intercepted arc with the given central angle measure in a circle with the given radius. Round to the nearest tenth. 29. , r = 4 yd SOLUTION: ANSWER: 5.2 yd 30. 105°, r = 18.2 cm SOLUTION: Method 1 Convert 105° to radian measure, and then use s = rθ to find the arc length. Substitute r = 18.2 and θ = . Method 2 Use s = to find the arc length. ANSWER: 33.4 cm Find the rotation in revolutions per minute given the angular speed and the radius given the linear speed and the rate of rotation. 36. = 104π rad/min SOLUTION: The angular speed is 104π radians per minute. Each revolution measures 2π radians 104π ÷ 2π = 52 The angle of rotation is 52 revolutions per minute. ANSWER: 52 rev/min 37. v = 82.3 m/s, 131 rev/min SOLUTION: 131 × 2π = 262π The linear speed is 82.3 meters per second with an angle of rotation of 262π radians per minute. Use the linear speed equation to find the radius, r. The radius is about 6 m. ANSWER: 6 m Find the area of each sector. 47. SOLUTION: The measure of the sector’s central angle θ is 177° and the radius is 18 feet. Convert the central angle measure to radians. Use the central angle and the radius to find the area of the sector. Therefore, the area of the sector is about 500.5 square feet. ANSWER: 2 500.5 ft 48. SOLUTION: The measure of the sector’s central angle θ is and the radius is 43.5 centimeters. Therefore, the area of the sector is about 247.7 square centimeters. ANSWER: 247.7 cm2 49. GAMES The dart board shown is divided into twenty equal sectors. If the diameter of the board is 18 inches, what area of the board does each sector cover? SOLUTION: 360° ÷ 20 or 18° If the board is divided into 20 equal sectors, then the central angle of each sector has a measure of 18°. So, the measure of a sector’s central angle is 18° and the radius is 9 inches. Convert the central angle measure to radians. Use the central angle and the radius to find the area of a sector. Therefore, each sector covers an area of about 12.7 square inches. ANSWER: 2 12.7 in 4-2 Degrees and Radians Write each degree measure in radians as a multiple of π and each radian measure in degrees. 10. 30° SOLUTION: To convert a degree measure to radians, multiply by ANSWER: 14. SOLUTION: To convert a radian measure to degrees, multiply by ANSWER: 120° Identify all angles that are coterminal with the given angle. Then find and draw one positive and one negative angle coterminal with the given angle. 20. 225° SOLUTION: All angles measuring are coterminal with a angle. Sample answer: Let n = 1 and −1. eSolutions Manual - Powered by Cognero Page 2 ANSWER: 225° + 360n°; Sample answer: 585°, −135° 24. SOLUTION: All angles measuring are coterminal with a angle. Sample answer: Let n = 1 and −1. ANSWER: Find the length of the intercepted arc with the given central angle measure in a circle with the given radius. Round to the nearest tenth. 29. , r = 4 yd SOLUTION: ANSWER: 5.2 yd 30. 105°, r = 18.2 cm SOLUTION: Method 1 Convert 105° to radian measure, and then use s = rθ to find the arc length. Substitute r = 18.2 and θ = . Method 2 Use s = to find the arc length. ANSWER: 33.4 cm Find the rotation in revolutions per minute given the angular speed and the radius given the linear speed and the rate of rotation. 36. = 104π rad/min SOLUTION: The angular speed is 104π radians per minute. Each revolution measures 2π radians 104π ÷ 2π = 52 The angle of rotation is 52 revolutions per minute. ANSWER: 52 rev/min 37. v = 82.3 m/s, 131 rev/min SOLUTION: 131 × 2π = 262π The linear speed is 82.3 meters per second with an angle of rotation of 262π radians per minute. Use the linear speed equation to find the radius, r. The radius is about 6 m. ANSWER: 6 m Find the area of each sector. 47. SOLUTION: The measure of the sector’s central angle θ is 177° and the radius is 18 feet. Convert the central angle measure to radians. Use the central angle and the radius to find the area of the sector. Therefore, the area of the sector is about 500.5 square feet. ANSWER: 2 500.5 ft 48. SOLUTION: The measure of the sector’s central angle θ is and the radius is 43.5 centimeters. Therefore, the area of the sector is about 247.7 square centimeters. ANSWER: 247.7 cm2 49. GAMES The dart board shown is divided into twenty equal sectors. If the diameter of the board is 18 inches, what area of the board does each sector cover? SOLUTION: 360° ÷ 20 or 18° If the board is divided into 20 equal sectors, then the central angle of each sector has a measure of 18°. So, the measure of a sector’s central angle is 18° and the radius is 9 inches. Convert the central angle measure to radians. Use the central angle and the radius to find the area of a sector. Therefore, each sector covers an area of about 12.7 square inches. ANSWER: 2 12.7 in 4-2 Degrees and Radians Write each degree measure in radians as a multiple of π and each radian measure in degrees. 10. 30° SOLUTION: To convert a degree measure to radians, multiply by ANSWER: 14. SOLUTION: To convert a radian measure to degrees, multiply by ANSWER: 120° Identify all angles that are coterminal with the given angle. Then find and draw one positive and one negative angle coterminal with the given angle. 20. 225° SOLUTION: All angles measuring are coterminal with a angle. Sample answer: Let n = 1 and −1. eSolutions Manual - Powered by Cognero Page 3 ANSWER: 225° + 360n°; Sample answer: 585°, −135° 24. SOLUTION: All angles measuring are coterminal with a angle. Sample answer: Let n = 1 and −1. ANSWER: Find the length of the intercepted arc with the given central angle measure in a circle with the given radius. Round to the nearest tenth. 29. , r = 4 yd SOLUTION: ANSWER: 5.2 yd 30. 105°, r = 18.2 cm SOLUTION: Method 1 Convert 105° to radian measure, and then use s = rθ to find the arc length. Substitute r = 18.2 and θ = . Method 2 Use s = to find the arc length. ANSWER: 33.4 cm Find the rotation in revolutions per minute given the angular speed and the radius given the linear speed and the rate of rotation. 36. = 104π rad/min SOLUTION: The angular speed is 104π radians per minute. Each revolution measures 2π radians 104π ÷ 2π = 52 The angle of rotation is 52 revolutions per minute. ANSWER: 52 rev/min 37. v = 82.3 m/s, 131 rev/min SOLUTION: 131 × 2π = 262π The linear speed is 82.3 meters per second with an angle of rotation of 262π radians per minute. Use the linear speed equation to find the radius, r. The radius is about 6 m. ANSWER: 6 m Find the area of each sector. 47. SOLUTION: The measure of the sector’s central angle θ is 177° and the radius is 18 feet. Convert the central angle measure to radians. Use the central angle and the radius to find the area of the sector. Therefore, the area of the sector is about 500.5 square feet. ANSWER: 2 500.5 ft 48. SOLUTION: The measure of the sector’s central angle θ is and the radius is 43.5 centimeters. Therefore, the area of the sector is about 247.7 square centimeters. ANSWER: 247.7 cm2 49. GAMES The dart board shown is divided into twenty equal sectors. If the diameter of the board is 18 inches, what area of the board does each sector cover? SOLUTION: 360° ÷ 20 or 18° If the board is divided into 20 equal sectors, then the central angle of each sector has a measure of 18°. So, the measure of a sector’s central angle is 18° and the radius is 9 inches. Convert the central angle measure to radians. Use the central angle and the radius to find the area of a sector. Therefore, each sector covers an area of about 12.7 square inches. ANSWER: 2 12.7 in 4-2 Degrees and Radians Write each degree measure in radians as a multiple of π and each radian measure in degrees. 10. 30° SOLUTION: To convert a degree measure to radians, multiply by ANSWER: 14. SOLUTION: To convert a radian measure to degrees, multiply by ANSWER: 120° Identify all angles that are coterminal with the given angle.
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