1. Convert degree measures of angles to radian measures, and vice versa.
You used the measures of acute angles in triangles given in degrees. (Lesson 4-1)
Convert degree measures of angles to radian measures, and vice versa.
Use angle measures to solve real-world problems.
Then/Now

2. vertex initial side terminal side standard position radian
coterminal angles
linear speed
angular speed
sector
Vocabulary

3. First, convert 0.125° into minutes and seconds.
Convert Between DMS and Decimal Degree Form
A. Write ° in DMS form.
First, convert 0.125° into minutes and seconds.
° = 329° + 1° = 60'
= 329° + 7.5' Simplify.
Next, convert 0.5' into seconds.
° = 329° + 7' + 1' = 60"
= 329° + 7' + 30" Simplify.
Therefore, ° can be written as 329°7'30".
Answer: 329°7'30"
Example 1

4. B. Write 35°12'7'' in decimal degree form to the nearest thousandth.
Convert Between DMS and Decimal Degree Form
B. Write 35°12'7'' in decimal degree form to the nearest thousandth.
Each minute is of a degree and each second is of a minute, so each second is of a degree.
35°12'7" = 35o + 12'
Example 1

5. Therefore, 35°12'7" can be written as about 35.202°.
Convert Between DMS and Decimal Degree Form
≈ 35° Simplify.
≈ ° Add.
Therefore, 35°12'7" can be written as about °.
Answer: °
Example 1

6. Write 141.275° in DMS form. A. 141°12'4.5" B. 141.2°45'0" C. 141°4'35"
Example 1

7. Key Concept 2

8. Key Concept 2

9. A. Write 135° in radians as a multiple of π.
Convert Between Degree and Radian Measure
A. Write 135° in radians as a multiple of π.
Answer:
Example 2

10. B. Write –30° in radians as a multiple of π.
Convert Between Degree and Radian Measure
B. Write –30° in radians as a multiple of π.
Answer:
Example 2

11. C. Write in degrees. = 120° Simplify. Answer: 120°
Convert Between Degree and Radian Measure
C. Write in degrees.
= 120° Simplify.
Answer: 120°
Example 2

12. D. Write in degrees. = 135° Simplify. Answer: –135°
Convert Between Degree and Radian Measure
D. Write in degrees.
= 135° Simplify.
Answer: –135°
Example 2

13. Write 150o in radians as a multiple of π.B. C. D.
Example 2

14. Key Concept 3

15. Find and Draw Coterminal Angles
A. Identify all angles that are coterminal with 80°. Then find and draw one positive and one negative angle coterminal with 80°.
All angles measuring 80° + 360n° are coterminal with an 80° angle. Let n = 1 and –1.
80° + 360(1)° = 80° + 360° or 440°
Example 3

16. Answer: 80o + 360no; Sample answers: 440o, –280o
Find and Draw Coterminal Angles
80° + 360(–1)° = 80° – 360° or –280°
Answer: 80o + 360no; Sample answers: 440o, –280o
Example 3

17. All angles measuring are coterminal with a angle. Let n = 1 and –1.
Find and Draw Coterminal Angles
B. Identify all angles that are coterminal with Then find and draw one positive and one negative angle coterminal with
All angles measuring are coterminal with a angle. Let n = 1 and –1.
Example 3

18. Answer: Sample answer:
Find and Draw Coterminal Angles
Answer: Sample answer:
Example 3

19. Identify one positive and one negative angle coterminal with a 126o angle.
B. 54°, –126°
C. 234°, –54°
D. 36°, –486°
Example 3

20. Key Concept 4

21. Find Arc Length
A. Find the length of the intercepted arc in a circle with a central angle measure of and a radius of 4 inches. Round to the nearest tenth.
Arc Length
r = 4 and
Simplify.
Example 4

22. The length of the intercepted arc is or about 4.2 inches.
Find Arc Length
The length of the intercepted arc is or about 4.2 inches.
Answer: 4.2 in.
Example 4

23. Find Arc Length
B. Find the length of the intercepted arc in a circle with a central angle measure of 125° and a radius of 7 centimeters. Round to the nearest tenth.
Method 1 Convert 125o to radian measure, and then use s = rθ to find the arc length.
Example 4

24. Substitute r = 7 and . Arc length s = r r = 7 and Simplify.
Find Arc Length
Substitute r = 7 and
Arc length
s = r
r = 7 and
Simplify.
Example 4

25. Method 2 Use to find the arc length.
Find Arc Length
Method 2 Use to find the arc length.
Arc length
r = 7 and θ = 125°
Simplify.
The length of the intercepted arc is or about 15.3 centimeters.
Answer: 15.3 cm
Example 4

26. Find the length of the intercepted arc in a circle with radius 6 centimeters and a central angle with measure .
A. 2.4 centimeters
B. 4.7 centimeters
C centimeters
D. 45°
Example 4

27. Key Concept 5

28. Find Angular and Linear Speeds
A. RECORDS A typical vinyl record has a diameter of 30 cm. When played on a turn table, the record spins at revolutions per minute. Find the angular speed, in radians per minute, of a record as it plays. Round to the nearest tenth.
Because each rotation measures 2π radians, revolutions correspond to an angle of rotation
Example 5

29. Answer: 209.4 radians per minute
Find Angular and Linear Speeds
Angular speed
Therefore, the angular speed of the record is or about radians per minute.
Answer: radians per minute
Example 5

30. A rotation of revolutions corresponds to an angle of rotation
Find Angular and Linear Speeds
B. RECORDS A typical vinyl record has a diameter of 30 cm. When played on a turn table, the record spins at revolutions per minute. Find the linear speed at the outer edge of the record as it spins, in centimeters per second.
A rotation of revolutions corresponds to an angle of rotation
Example 5

31. Linear Speed s = r minute Simplify. Find Angular and Linear Speeds
Example 5

32. Find Angular and Linear Speeds
Use dimensional analysis to convert this speed from centimeters per minute to centimeters per second.
Therefore, the linear speed of the record is about 52.4 centimeters per second.
Answer: about 52.4 cm/s
Example 5

33. CAROUSEL Find the angular speed of a carousel in radians per minute if the diameter is 6 feet and it rotates at a rate of 10 revolutions per minute.
A radians per minute
B radians per minute
C radians per minute
D radians per minute
Example 5