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

4-2 Degrees and Radians ANSWER: 120° Write each degree measure in radians as a Identify all angles that are coterminal with the multiple of π and each radian measure in given angle. Then find and draw one positive degrees. and one negative angle coterminal with the 10. 30° given angle. 20. 225° SOLUTION: To convert a degree measure to radians, multiply by SOLUTION: All angles measuring are coterminal with a angle.

Sample answer: Let n = 1 and −1.

ANSWER:

14.

SOLUTION: To convert a radian measure to degrees, multiply by

ANSWER: 225° + 360n°; Sample answer: 585°, −135°

ANSWER:

120°

Identify all angles that are coterminal with the 24. given angle. Then find and draw one positive and one negative angle coterminal with the given angle. SOLUTION: 20. 225° All angles measuring are coterminal with SOLUTION:

All angles measuring are coterminal a angle. with a angle. Sample answer: Let n = 1 and −1. Sample answer: Let n = 1 and −1. eSolutions Manual - Powered by Cognero Page 1

ANSWER: 225° + 360n°; Sample answer: 585°, −135°

ANSWER:

24.

SOLUTION:

All angles measuring are coterminal with 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. a angle.

29. , r = 4 yd Sample answer: Let n = 1 and −1. 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 = . ANSWER:

Method 2

Use s = to find the arc length. 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: 33.4 cm Find the rotation in revolutions per minute given the angular speed and the radius given ANSWER: the linear speed and the rate of rotation. 5.2 yd 36. = 104π rad/min SOLUTION: 30. 105°, r = 18.2 cm The angular speed is 104π radians per minute. SOLUTION: Method 1 Convert 105° to radian measure, and then use s = rθ to find the arc length.

Each revolution measures 2π radians

104π ÷ 2π = 52

Substitute r = 18.2 and θ = . 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 Method 2

Use s = to find the arc length. 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.

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. The radius is about 6 m. ANSWER: 6 m Find the area of each sector.

Each revolution measures 2π radians

104π ÷ 2π = 52 47. The angle of rotation is 52 revolutions per minute. SOLUTION: ANSWER: The measure of the sector’s central angle θ is 177° 52 rev/min and the radius is 18 feet. Convert the central angle measure to radians.

37. v = 82.3 m/s, 131 rev/min SOLUTION: 131 × 2π = 262π Use the central angle and the radius to find the area The linear speed is 82.3 meters per second with an of the sector. angle of rotation of 262π radians per minute.

Use the linear speed equation to find the radius, r.

Therefore, the area of the sector is about 500.5 square feet. ANSWER: 2 500.5 ft

48. The radius is about 6 m. SOLUTION: ANSWER: The measure of the sector’s central angle θ is 6 m and the radius is 43.5 centimeters. Find the area of each sector.

47. Therefore, the area of the sector is about 247.7 SOLUTION: square centimeters. The measure of the sector’s central angle θ is 177° and the radius is 18 feet. Convert the central angle ANSWER: measure to radians. 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? Use the central angle and the radius to find the area of the sector.

Therefore, the area of the sector is about 500.5 SOLUTION: square feet. 360° ÷ 20 or 18° If the board is divided into 20 equal sectors, then the ANSWER: central angle of each sector has a measure of 18°. 2 500.5 ft So, the measure of a sector’s central angle is 18° and the radius is 9 inches.

Convert the central angle measure to radians.

48.

SOLUTION: Use the central angle and the radius to find the area The measure of the sector’s central angle θ is of a sector.

and the radius is 43.5 centimeters.

Therefore, each sector covers an area of about 12.7 square inches. Therefore, the area of the sector is about 247.7 ANSWER: square centimeters. 2 12.7 in 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 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° Write each degree measure in radians as a multiple of π and each radian measure in SOLUTION: degrees. All angles measuring are coterminal 10. 30° with a angle. SOLUTION: To convert a degree measure to radians, multiply by Sample answer: Let n = 1 and −1.

ANSWER:

14.

SOLUTION: To convert a radian measure to degrees, multiply by ANSWER: 225° + 360n°; Sample answer: 585°, −135°

24. ANSWER: 120° SOLUTION: Identify all angles that are coterminal with the All angles measuring are coterminal with given angle. Then find and draw one positive

and one negative angle coterminal with the a angle. given angle. 20. 225° Sample answer: Let n = 1 and −1. SOLUTION: All angles measuring are coterminal with a angle.

Sample answer: Let n = 1 and −1.

ANSWER: ANSWER: 225° + 360n°; Sample answer: 585°, −135°

4-2 Degrees and Radians

Find the length of the intercepted arc with the 24. given central angle measure in a circle with the given radius. Round to the nearest tenth. SOLUTION: 29. , r = 4 yd All angles measuring are coterminal with SOLUTION:

a angle.

Sample answer: Let n = 1 and −1.

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 = .

ANSWER:

Method 2

Use s = to find the arc length.

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 ANSWER: SOLUTION: 33.4 cm eSolutions Manual - Powered by Cognero Page 2 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: ANSWER: The angular speed is 104π radians per minute. 5.2 yd

30. 105°, r = 18.2 cm SOLUTION:

Method 1 Each revolution measures 2π radians Convert 105° to radian measure, and then use s = rθ to find the arc length. 104π ÷ 2π = 52

The angle of rotation is 52 revolutions per minute. ANSWER: 52 rev/min Substitute r = 18.2 and θ = . 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.

Method 2 Use the linear speed equation to find the radius, r.

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 radius is about 6 m. the linear speed and the rate of rotation. 36. = 104π rad/min ANSWER: SOLUTION: 6 m The angular speed is 104π radians per minute. Find the area of each sector.

47.

Each revolution measures 2π radians SOLUTION: The measure of the sector’s central angle θ is 177° 104π ÷ 2π = 52 and the radius is 18 feet. Convert the central angle measure to radians. The angle of rotation is 52 revolutions per minute. ANSWER: 52 rev/min

Use the central angle and the radius to find the area 37. v = 82.3 m/s, 131 rev/min of the sector. SOLUTION: 131 × 2π = 262π

The linear speed is 82.3 meters per second with an angle of rotation of 262π radians per minute. Therefore, the area of the sector is about 500.5 Use the linear speed equation to find the radius, r. 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. The radius is about 6 m.

ANSWER: 6 m Find the area of each sector.

Therefore, the area of the sector is about 247.7 square centimeters. ANSWER: 47. 247.7 cm2 SOLUTION: 49. GAMES The dart board shown is divided into The measure of the sector’s central angle θ is 177° twenty equal sectors. If the diameter of the board is and the radius is 18 feet. Convert the central angle 18 inches, what area of the board does each sector measure to radians. cover?

Use the central angle and the radius to find the area of the sector.

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°. Therefore, the area of the sector is about 500.5 So, the measure of a sector’s central angle is 18° square feet. and the radius is 9 inches.

ANSWER: Convert the central angle measure to radians. 2 500.5 ft

Use the central angle and the radius to find the area of a sector.

48. SOLUTION: The measure of the sector’s central angle θ is

and the radius is 43.5 centimeters. Therefore, each sector covers an area of about 12.7

square inches. ANSWER: 2 12.7 in

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?

Convert the central angle measure to radians.

Use the central angle and the radius to find the area of a sector.

ANSWER:

14.

SOLUTION: To convert a radian measure to degrees, multiply by

Write each degree measure in radians as a multiple of π and each radian measure in degrees. ANSWER: 10. 30° 120° SOLUTION: Identify all angles that are coterminal with the To convert a degree measure to radians, multiply by 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. ANSWER:

14.

SOLUTION:

To convert a radian measure to degrees, multiply by

ANSWER: ANSWER: 225° + 360n°; Sample answer: 585°, −135° 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: 24. All angles measuring are coterminal with a angle. SOLUTION:

Sample answer: Let n = 1 and −1. All angles measuring are coterminal with

a angle.

Sample answer: Let n = 1 and −1.

ANSWER: 225° + 360n°; Sample answer: 585°, −135°

ANSWER:

24.

SOLUTION:

All angles measuring are coterminal with

a angle.

Sample answer: Let n = 1 and −1. 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.

ANSWER:

Substitute r = 18.2 and θ = .

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. Method 2 29. , r = 4 yd Use s = to find the arc length. SOLUTION:

ANSWER: ANSWER: 4-2 5.2Degrees yd and Radians 33.4 cm

30. 105°, r = 18.2 cm Find the rotation in revolutions per minute given the angular speed and the radius given SOLUTION: the linear speed and the rate of rotation. Method 1 36. = 104π rad/min Convert 105° to radian measure, and then use s = rθ to find the arc length. SOLUTION: The angular speed is 104π radians per minute.

Substitute r = 18.2 and θ = .

Each revolution measures 2π radians

104π ÷ 2π = 52

The angle of rotation is 52 revolutions per minute. ANSWER:

52 rev/min Method 2

Use s = to find the arc length. 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. 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.

The radius is about 6 m.

Each revolution measures 2π radians ANSWER: 6 m 104π ÷ 2π = 52 eSolutions Manual - Powered by Cognero Find the area of each sector. Page 3 The angle of rotation is 52 revolutions per minute. ANSWER: 52 rev/min 47. 37. v = 82.3 m/s, 131 rev/min SOLUTION: SOLUTION: The measure of the sector’s central angle θ is 177° 131 × 2π = 262π and the radius is 18 feet. Convert the central angle measure to radians. 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. 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

The radius is about 6 m. ANSWER: 6 m Find the area of each sector. 48. SOLUTION: The measure of the sector’s central angle θ is

and the radius is 43.5 centimeters. 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. Therefore, the area of the sector is about 247.7 square centimeters. ANSWER: 2 Use the central angle and the radius to find the area 247.7 cm of the sector. 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?

Therefore, the area of the sector is about 500.5 square feet. ANSWER: 2 500.5 ft 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. 48. SOLUTION: Convert the central angle measure to radians.

The measure of the sector’s central angle θ is

and the radius is 43.5 centimeters.

Use the central angle and the radius to find the area of a sector.

Therefore, the area of the sector is about 247.7 square centimeters. ANSWER: Therefore, each sector covers an area of about 12.7 square inches. 247.7 cm2 ANSWER: 49. GAMES The dart board shown is divided into 2 twenty equal sectors. If the diameter of the board is 12.7 in 18 inches, what area of the board does each sector cover?

Convert the central angle measure to radians.

Use the central angle and the radius to find the area of a sector.

ANSWER:

14.

SOLUTION: To convert a radian measure to degrees, multiply by

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: 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° ANSWER: SOLUTION: All angles measuring are coterminal with a angle.

Sample answer: Let n = 1 and −1.

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 ANSWER: and one negative angle coterminal with the 225° + 360n°; Sample answer: 585°, −135° given angle. 20. 225° SOLUTION: All angles measuring are coterminal with a angle.

Sample answer: Let n = 1 and −1. 24.

SOLUTION:

All angles measuring are coterminal with

a angle.

Sample answer: Let n = 1 and −1.

ANSWER:

225° + 360n°; Sample answer: 585°, −135°

24.

SOLUTION: ANSWER: All angles measuring are coterminal with

a angle.

Sample answer: Let n = 1 and −1.

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: ANSWER: Method 1 Convert 105° to radian measure, and then use s = rθ to find the arc length.

Substitute r = 18.2 and θ = . 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:

Method 2

Use s = to find the arc length.

ANSWER: 5.2 yd

30. 105°, r = 18.2 cm SOLUTION: ANSWER: Method 1 33.4 cm Convert 105° to radian measure, and then use s = rθ to find the arc length. 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. Substitute r = 18.2 and θ = .

Each revolution measures 2π radians

104π ÷ 2π = 52 Method 2

Use s = to find the arc length. 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

ANSWER: The linear speed is 82.3 meters per second with an 33.4 cm angle of rotation of 262π radians per minute.

Find the rotation in revolutions per minute Use the linear speed equation to find the radius, r. 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 radius is about 6 m. The angle of rotation is 52 revolutions per minute. ANSWER: ANSWER: 6 m 52 rev/min Find the area of each sector.

37. v = 82.3 m/s, 131 rev/min SOLUTION: 131 × 2π = 262π 47. The linear speed is 82.3 meters per second with an SOLUTION: angle of rotation of 262π radians per minute.

The measure of the sector’s central angle θ is 177° and the radius is 18 feet. Convert the central angle Use the linear speed equation to find the radius, r. 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. The radius is about 6 m. ANSWER: ANSWER: 2 4-2 Degrees and Radians 500.5 ft 6 m Find the area of each sector.

48. 47. SOLUTION: SOLUTION: The measure of the sector’s central angle θ is The measure of the sector’s central angle θ is 177° and the radius is 18 feet. Convert the central angle and the radius is 43.5 centimeters. measure to radians.

Use the central angle and the radius to find the area Therefore, the area of the sector is about 247.7 of the sector. square centimeters.

ANSWER: 247.7 cm2

49. GAMES The dart board shown is divided into Therefore, the area of the sector is about 500.5 twenty equal sectors. If the diameter of the board is square feet. 18 inches, what area of the board does each sector cover? ANSWER: 2 500.5 ft

SOLUTION: 48. 360° ÷ 20 or 18° SOLUTION: If the board is divided into 20 equal sectors, then the central angle of each sector has a measure of 18°. The measure of the sector’s central angle θ is So, the measure of a sector’s central angle is 18°

and the radius is 43.5 centimeters. and the radius is 9 inches.

Convert the central angle measure to radians.

Therefore, the area of the sector is about 247.7 Use the central angle and the radius to find the area square centimeters. of a sector.

ANSWER: 2 eSolutions247.7Manual cm - Powered by Cognero Page 4

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 Therefore, each sector covers an area of about 12.7

cover? square inches. ANSWER: 2 12.7 in

Convert the central angle measure to radians.

Use the central angle and the radius to find the area of a sector.

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.

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: 4-2 Degrees and Radians 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?

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