1. Splash Screen

2. Five-Minute Check (over Lesson 12–1) CCSS Then/Now New Vocabulary
Key Concept: Angle Measures
Example 1: Draw an Angle in Standard Position
Example 2: Real-World Example: Draw an Angle in Standard Position
Example 3: Find Coterminal Angles
Key Concept: Convert Between Degrees and Radians
Example 4: Convert Between Degrees and Radians
Concept Summary: Degrees and Radians
Key Concept: Arc Length
Example 5: Real-World Example: Find Arc Length
Lesson Menu

3. Find sin , cos , and tan . A. B. C. D.
5-Minute Check 1

4. Find sin , cos , and tan . A. B. C. D.
5-Minute Check 1

5. Find csc , sec , and cot . A. B. C. D.
5-Minute Check 2

6. Find csc , sec , and cot . A. B. C. D.
5-Minute Check 2

7. Find the value of a.
A. 13.9
B. 12.3
C. 6.9
D. 4.5
5-Minute Check 3

8. Find the value of a.
A. 13.9
B. 12.3
C. 6.9
D. 4.5
5-Minute Check 3

9. Find the measure of B.
A. 30°
B. 45°
C. 60°
D. 90°
5-Minute Check 4

10. Find the measure of B.
A. 30°
B. 45°
C. 60°
D. 90°
5-Minute Check 4

11. Find the value of c.
A. 13.9
B. 12.3
C. 9.1
D. 6.9
5-Minute Check 5

12. Find the value of c.
A. 13.9
B. 12.3
C. 9.1
D. 6.9
5-Minute Check 5

13. David needs a ramp that rises to a height of 3 feet at a 15° angle
David needs a ramp that rises to a height of 3 feet at a 15° angle. Write an equation for the length ℓ of the ramp.
A. ℓ = 3 sin 15°
B. ℓ = 3 cos 15°
C. ℓ
D. ℓ
5-Minute Check 6

14. David needs a ramp that rises to a height of 3 feet at a 15° angle
David needs a ramp that rises to a height of 3 feet at a 15° angle. Write an equation for the length ℓ of the ramp.
A. ℓ = 3 sin 15°
B. ℓ = 3 cos 15°
C. ℓ
D. ℓ
5-Minute Check 6

15. At a construction site, the workers need to build a ramp up to the second story of a house. The angle of inclination of the ramp cannot be more than 20°. Find the length of the ramp if the distance to the second story is 15 feet.
A ft
B ft
C ft
D ft
5-Minute Check 7

16. At a construction site, the workers need to build a ramp up to the second story of a house. The angle of inclination of the ramp cannot be more than 20°. Find the length of the ramp if the distance to the second story is 15 feet.
A ft
B ft
C ft
D ft
5-Minute Check 7

17. Mathematical Practices 2 Reason abstractly and quantitatively.
Content Standards
F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Mathematical Practices
2 Reason abstractly and quantitatively.
CCSS

18. You used angles with degree measures.
Draw and find angles in standard position.
Convert between degree measures and radian measures.
Then/Now

19. standard position initial side terminal side coterminal angles radian
central angle
arc length
Vocabulary

20. Concept

21. A. Draw an angle with a measure of 210° in standard position.
Draw an Angle in Standard Position
A. Draw an angle with a measure of 210° in standard position.
210° = 180° + 30°
Draw the terminal side of the angle 30° counterclockwise past the negative x-axis.
Answer:
Example 1

22. A. Draw an angle with a measure of 210° in standard position.
Draw an Angle in Standard Position
A. Draw an angle with a measure of 210° in standard position.
210° = 180° + 30°
Draw the terminal side of the angle 30° counterclockwise past the negative x-axis.
Answer:
Example 1

23. B. Draw an angle with a measure of –45° in standard position.
Draw an Angle in Standard Position
B. Draw an angle with a measure of –45° in standard position.
The angle is negative. Draw the terminal side 45° clockwise from the positive x-axis.
Answer:
Example 1

24. B. Draw an angle with a measure of –45° in standard position.
Draw an Angle in Standard Position
B. Draw an angle with a measure of –45° in standard position.
The angle is negative. Draw the terminal side 45° clockwise from the positive x-axis.
Answer:
Example 1

25. A. Draw an angle with a measure of 225° in standard position.
A. B.
C. D.
Example 1

26. A. Draw an angle with a measure of 225° in standard position.
A. B.
C. D.
Example 1

27. B. Draw an angle with a measure of –60° in standard position.
A. B.
C. D.
Example 1

28. B. Draw an angle with a measure of –60° in standard position.
A. B.
C. D.
Example 1

29. Draw the terminal side of the angle 180° past the positive x-axis.
Draw an Angle in Standard Position
A. DIVING In a springboard diving competition, a diver made a 900-degree rotation before slicing into the water. Draw an angle in standard position that measures 900°.
900° = 360° + 360° + 180°
Draw the terminal side of the angle 180° past the positive x-axis.
Answer:
Example 2

30. Draw the terminal side of the angle 180° past the positive x-axis.
Draw an Angle in Standard Position
A. DIVING In a springboard diving competition, a diver made a 900-degree rotation before slicing into the water. Draw an angle in standard position that measures 900°.
900° = 360° + 360° + 180°
Draw the terminal side of the angle 180° past the positive x-axis.
Answer:
Example 2

31. SNOWBOARDING While riding down the mountain, a snowboarder goes off a jump and turns 600° before touching down onto the snow again. Determine how many degrees past the positive x-axis the snowboarder lands.
A. 120°
B. 180°
C. 240°
D. 300°
Example 2

32. SNOWBOARDING While riding down the mountain, a snowboarder goes off a jump and turns 600° before touching down onto the snow again. Determine how many degrees past the positive x-axis the snowboarder lands.
A. 120°
B. 180°
C. 240°
D. 300°
Example 2

33. negative angle: 210° – 360° = –150°
Find Coterminal Angles
A. Find an angle with a positive measure and an angle with a negative measure that are coterminal with 210°.
positive angle: 210° + 360° = 570°
negative angle: 210° – 360° = –150°
Answer:
Example 3

34. negative angle: 210° – 360° = –150°
Find Coterminal Angles
A. Find an angle with a positive measure and an angle with a negative measure that are coterminal with 210°.
positive angle: 210° + 360° = 570°
negative angle: 210° – 360° = –150°
Answer: 570° and –150°
Example 3

35. negative angle: 120° – 360° = –480°
Find Coterminal Angles
B. Find an angle with a positive measure and an angle with a negative measure that are coterminal with –120°.
positive angle: 120° + 360° = 240°
negative angle: 120° – 360° = –480°
Answer:
Example 3

36. negative angle: 120° – 360° = –480°
Find Coterminal Angles
B. Find an angle with a positive measure and an angle with a negative measure that are coterminal with –120°.
positive angle: 120° + 360° = 240°
negative angle: 120° – 360° = –480°
Answer: 240° and –480°
Example 3

37. A. Find an angle with a positive measure and an angle with a negative measure that are coterminal with 330°.
A. –30°, 690°
B. –30°, 630°
C. –60°, 630°
D. –60°, 720°
Example 3

38. A. Find an angle with a positive measure and an angle with a negative measure that are coterminal with 330°.
A. –30°, 690°
B. –30°, 630°
C. –60°, 630°
D. –60°, 720°
Example 3

39. B. Find an angle with a positive measure and an angle with a negative measure that are coterminal with –80°.
A. –380°, 220°
B. –440°, 280°
C. –320°, 380°
D. –400°, 300°
Example 3

40. B. Find an angle with a positive measure and an angle with a negative measure that are coterminal with –80°.
A. –380°, 220°
B. –440°, 280°
C. –320°, 380°
D. –400°, 300°
Example 3

41. Concept

42. A. Rewrite 30° in radians. Answer: Convert Between Degrees and Radians
Example 4

43. A. Rewrite 30° in radians. Answer: Convert Between Degrees and Radians
Example 4

44. B. Rewrite in degrees. Answer: Convert Between Degrees and Radians
Example 4

45. B. Rewrite in degrees. Answer: –300°
Convert Between Degrees and Radians
B. Rewrite in degrees.
Answer: –300°
Example 4

46. A. Rewrite 45° in radians. A. B. C. D.
Example 4

47. A. Rewrite 45° in radians. A. B. C. D.
Example 4

48. B. Rewrite in degrees.
A. 70°
B. 80°
C. 30°
D. 60°
Example 4

49. B. Rewrite in degrees.
A. 70°
B. 80°
C. 30°
D. 60°
Example 4

50. Concept

51. Concept

52. Step 1 Find the central angle in radians.
Find Arc Length
TRUCKS The steering wheel on a monster truck has a radius of 11 inches. How far does a point on the steering wheel travel if the wheel makes four fifths of a rotation?
Step 1 Find the central angle in radians. 
Example 5

53. Step 2 Use the radius and the central angle to find the arc length.
Find Arc Length
Step 2 Use the radius and the central angle to find the arc length.
s = r Write the formula for arc length.
≈ 55.3 in. Use a calculator to simplify.
Answer:
Example 5

54. Step 2 Use the radius and the central angle to find the arc length.
Find Arc Length
Step 2 Use the radius and the central angle to find the arc length.
s = r Write the formula for arc length.
≈ 55.3 in. Use a calculator to simplify.
Answer: A point on the steering wheel will travel about 55.3 inches after four fifths of a rotation.
Example 5

55. BOATS The steering wheel on a yacht has a radius of 16 inches
BOATS The steering wheel on a yacht has a radius of 16 inches. How far does a point on the steering wheel travel if the wheel makes five sevenths of a rotation?
A in.
B in.
C in.
D in.
Example 5

56. BOATS The steering wheel on a yacht has a radius of 16 inches
BOATS The steering wheel on a yacht has a radius of 16 inches. How far does a point on the steering wheel travel if the wheel makes five sevenths of a rotation?
A in.
B in.
C in.
D in.
Example 5

57. End of the Lesson