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Splash Screen. Five-Minute Check (over Lesson 12–5) CCSS Then/Now New Vocabulary Key Concept: Functions on a Unit Circle Example 1: Find Sine and Cosine Given a Point on the Unit Circle Example 2: Identify the Period Example 3: Real-World Example: Use Trigonometric Functions

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 12–5) CCSS Then/Now New Vocabulary Key Concept: Functions on a Unit Circle Example 1: Find Sine and Cosine Given a Point on the Unit Circle Example 2: Identify the Period Example 3: Real-World Example: Use Trigonometric Functions Example 4: Evaluate Trigonometric Functions Lesson Menu

  3. Determine whether the triangle should be solved by beginning with the Law of Sines or Law of Cosines.Then solve the triangle. a = 7, b = 6, c = 5 A. Law of Sines; A = 79°, B = 57°, C = 44° B. Law of Cosines; A = 79°, B = 57°, C = 44° C. Law of Sines; A = 57°, B = 79°, C = 44° D. Law of Cosines; A = 57°, B = 79°, C = 44° 5-Minute Check 1

  4. Determine whether the triangle should be solved by beginning with the Law of Sines or Law of Cosines.Then solve the triangle. a = 7, b = 6, c = 5 A. Law of Sines; A = 79°, B = 57°, C = 44° B. Law of Cosines; A = 79°, B = 57°, C = 44° C. Law of Sines; A = 57°, B = 79°, C = 44° D. Law of Cosines; A = 57°, B = 79°, C = 44° 5-Minute Check 1

  5. Determine whether the triangle should be solved by beginning with the Law of Sines or Law of Cosines.Then solve the triangle. A = 89°, a = 14, b = 9 A.Law of Sines; B = 40°, C = 51°, c = 10.9 B.Law of Cosines; B = 40°, C = 51°, c = 10.9 C.Law of Sines; B = 51°, C = 40°, c = 10.9 D.Law of Cosines; B = 51°, C = 40°, c = 10.9 5-Minute Check 2

  6. Determine whether the triangle should be solved by beginning with the Law of Sines or Law of Cosines.Then solve the triangle. A = 89°, a = 14, b = 9 A.Law of Sines; B = 40°, C = 51°, c = 10.9 B.Law of Cosines; B = 40°, C = 51°, c = 10.9 C.Law of Sines; B = 51°, C = 40°, c = 10.9 D.Law of Cosines; B = 51°, C = 40°, c = 10.9 5-Minute Check 2

  7. The arms on a pair of tongs are each 8 inches long. They can open to an angle of up to 120°. What is the width of the largest object that can be grasped using these tongs? A. 4 in. B. 6.2 in. C. 13.9 in. D. 15.3 in. 5-Minute Check 3

  8. The arms on a pair of tongs are each 8 inches long. They can open to an angle of up to 120°. What is the width of the largest object that can be grasped using these tongs? A. 4 in. B. 6.2 in. C. 13.9 in. D. 15.3 in. 5-Minute Check 3

  9. What information will allow you to use the Law of Cosines to solve a triangle? A. two sides of a triangle and the included angle B. three angles C. two angles and any side D. two sides and the angle opposite one of the sides 5-Minute Check 4

  10. What information will allow you to use the Law of Cosines to solve a triangle? A. two sides of a triangle and the included angle B. three angles C. two angles and any side D. two sides and the angle opposite one of the sides 5-Minute Check 4

  11. 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. F.TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. Mathematical Practices 7 Look for and make use of structure. CCSS

  12. You evaluated trigonometric functions using reference angles. • Find values of trigonometric functions based on the unit circle. • Use the properties of periodic functions to evaluate trigonometric functions. Then/Now

  13. unit circle • circular function • periodic function • cycle • period Vocabulary

  14. Concept

  15. The terminal side of angle  in standard position intersects the unit circle at . Find cos  and sin . , Find Sine and Cosine Given a Point on the Unit Circle Example 1

  16. The terminal side of angle  in standard position intersects the unit circle at . Find cos  and sin . , Find Sine and Cosine Given a Point on the Unit Circle Example 1

  17. The terminal side of angle  in standard position intersects the unit circle at Find cos  and sin . A. B. C. D. Example 1

  18. The terminal side of angle  in standard position intersects the unit circle at Find cos  and sin . A. B. C. D. Example 1

  19. The pattern repeats at and so on. Identify the Period Determine the period of the function. Example 2

  20. The pattern repeats at and so on. Identify the Period Determine the period of the function. Example 2

  21. A. B. C. D. Determine the period of the function. Example 2

  22. A. B. C. D. Determine the period of the function. Example 2

  23. Use Trigonometric Functions A. CYCLING The pedals on a bicycle rotate as the bike is being ridden. The height of a bicycle pedal varies periodically as a function of the time, as shown in the figure. Notice that the pedal makes one complete rotation every two seconds. Make a table showing the height of a bicycle pedal at 3.0, 3.5, 4.0, 4.5, and 5.0 seconds. Example 3

  24. Use Trigonometric Functions At 3.0 seconds the pedal is in the same location as 3.0 – 2.0 or 1 second, which is 4 inches high. At 3.5 seconds the pedal is 11 inches high. At 4.0 seconds the pedal is 18 inches high. At 4.5 seconds the pedal is 11 inches high. At 5.0 seconds, the pedal is at 4 inches high. Answer: Example 3

  25. Use Trigonometric Functions At 3.0 seconds the pedal is in the same location as 3.0 – 2.0 or 1 second, which is 4 inches high. At 3.5 seconds the pedal is 11 inches high. At 4.0 seconds the pedal is 18 inches high. At 4.5 seconds the pedal is 11 inches high. At 5.0 seconds, the pedal is at 4 inches high. Answer: Example 3

  26. Use Trigonometric Functions B. CYCLING Identify the period of the function. The period is the time it takes to complete one rotation. Answer: Example 3

  27. Use Trigonometric Functions B. CYCLING Identify the period of the function. The period is the time it takes to complete one rotation. Answer: So, the period is 2 seconds. Example 3

  28. Use Trigonometric Functions C. CYCLING Graph the function. Let the horizontal axis represent the time t and the vertical axis represent the height h in inches that the pedal is from the ground. The maximum height of the pedal is 18 inches, and the minimum height is 4 inches. Because the period of the function is 2 seconds, the pattern of the graph repeats in intervals of 2 seconds, such as from 3.0 seconds to 5.0 seconds. Example 3

  29. Use Trigonometric Functions Answer: Example 3

  30. Use Trigonometric Functions Answer: Example 3

  31. A. STUNT CYCLING The pedals on a trick oversized bicycle rotate as the bike is being ridden. The height of a bicycle pedal varies periodically as a function of the time, as shown in the figure. Notice that the pedal makes one complete rotation every 4.0 seconds. Choose the correct table showing the height of a bicycle pedal at 0.0, 1.0, 2.0, 3.0, and 4.0 seconds. Example 3

  32. A.B. C.D. Example 3

  33. A.B. C.D. Example 3

  34. B. Identify the period of the function. A. 1.0 second B. 2.0 seconds C. 4.0 seconds D. 8.0 seconds Example 3

  35. B. Identify the period of the function. A. 1.0 second B. 2.0 seconds C. 4.0 seconds D. 8.0 seconds Example 3

  36. C. Choose the correct maximum height and minimum height of the graph of the function if the horizontal axis represents the time t and the vertical axis represents the height in inches that the pedal is from the ground. A. maximum height = 20 in. and minumum height = 10 in. B. maximum height = 40 in. and minumum height = 0 in. C. maximum height = 40 in. and minumum height = 30 in. D. maximum height = 30 in. and minumum height = 10 in. Example 3

  37. C. Choose the correct maximum height and minimum height of the graph of the function if the horizontal axis represents the time t and the vertical axis represents the height in inches that the pedal is from the ground. A. maximum height = 20 in. and minumum height = 10 in. B. maximum height = 40 in. and minumum height = 0 in. C. maximum height = 40 in. and minumum height = 30 in. D. maximum height = 30 in. and minumum height = 10 in. Example 3

  38. Evaluate Trigonometric Functions A. Find the exact value of cos 690°. cos 690° = cos(330° + 360°) = cos 330° Example 4

  39. Evaluate Trigonometric Functions A. Find the exact value of cos 690°. cos 690° = cos(330° + 360°) = cos 330° Example 4

  40. B. Find the exact value of Evaluate Trigonometric Functions Example 4

  41. B. Find the exact value of Evaluate Trigonometric Functions Example 4

  42. A. B. C. D. A. Find the exact value of cos 660°. Example 4

  43. A. B. C. D. A. Find the exact value of cos 660°. Example 4

  44. B. Find the exact value of A. B. C. D. Example 4

  45. B. Find the exact value of A. B. C. D. Example 4

  46. End of the Lesson

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