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Explore angle measures, convert between degrees and radians, find coterminal angles, and more in this comprehensive lesson. Understand standard position, arc length, and practical real-world examples.

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

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

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

  6. A. B. C. D. Find csc , sec , and cot . 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. A.ℓ = 3 sin 15° B.ℓ = 3 cos 15° C.ℓ D.ℓ 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. 5-Minute Check 6

  14. A.ℓ = 3 sin 15° B.ℓ = 3 cos 15° C.ℓ D.ℓ 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. 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. 5.13 ft B. 15.96 ft C. 41.21 ft D. 43.86 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. 5.13 ft B. 15.96 ft C. 41.21 ft D. 43.86 ft 5-Minute Check 7

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

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

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

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

  29. 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°. Answer: 900° = 360° + 360° + 180° Draw the terminal side of the angle 180° past the positive x-axis. Example 2

  30. 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°. Answer: 900° = 360° + 360° + 180° Draw the terminal side of the angle 180° past the positive x-axis. 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. 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. 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. 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. 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. Convert Between Degrees and Radians A. Rewrite 30° in radians. Answer: Example 4

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

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

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

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

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

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