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Splash Screen. Five-Minute Check (over Lesson 4-5) Then/Now New Vocabulary Example 1: Evaluate Inverse Sine Functions Example 2: Evaluate Inverse Cosine Functions Example 3: Evaluate Inverse Tangent Functions Key Concept: Inverse Trigonometric Functions

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 4-5) Then/Now New Vocabulary Example 1: Evaluate Inverse Sine Functions Example 2: Evaluate Inverse Cosine Functions Example 3: Evaluate Inverse Tangent Functions Key Concept: Inverse Trigonometric Functions Example 4: Sketch Graphs of Inverse Trigonometric Functions Example 5: Real-World Example: Use an Inverse Trigonometric Function Key Concept: Domain of Compositions of Trigonometric Functions Example 6: Use Inverse Trigonometric Properties Example 7: Evaluate Compositions of Trigonometric Functions Example 8: Evaluate Compositions of Trigonometric Functions Lesson Menu

  3. A.x = nπ, where n is an integer B. , where n is an odd integer C., where n is an integer D.x = nπ, where n is an odd integer A. Locate the vertical asymptotes of y = 2 sec x. 5–Minute Check 1

  4. A. B. C. D. B. Sketch the graph of y = 2 sec x. 5–Minute Check 1

  5. A.The damping factor is . B.The damping factor is – . C.The damping factor is 3x. D.The damping factor is 3. A. Identify the damping factor f(x) of y = 3x sin x. 5–Minute Check 2

  6. A. B. C. D. B. Use a graphing calculator to sketch thegraphs of y = 3x sin x, the damping factor f(x) of y = 3x sin x, and –f(x) in the same viewing window. 5–Minute Check 2

  7. C. Describe the graph of y = 3x sin x. A.The amplitude of the function is increasing as x approaches the origin. B.The amplitude of the function is decreasing as x approaches the origin. C. The amplitude oscillates between f(x) = x2 and f(x) = –x2. D. The amplitude is 2. 5–Minute Check 2

  8. A. B. C. D. Write an equation for a secant function with aperiod of 5π , a phase shift of –2π, and a verticalshift of –3. 5–Minute Check 3

  9. You found and graphed the inverses of relations and functions. (Lesson 1-7) • Evaluate and graph inverse trigonometric functions. • Find compositions of trigonometric functions. Then/Now

  10. arcsine function • arccosine function • arctangent function Vocabulary

  11. A. Find the exact value of , if it exists. Find a point on the unit circle on the interval with a y-coordinate of . When t = Therefore, Evaluate Inverse Sine Functions Example 1

  12. Answer: Check If  Evaluate Inverse Sine Functions Example 1

  13. B. Find the exact value of , if it exists. Find a point on the unit circle on the interval with a y-coordinate of When t = , sin t = Therefore, arcsin Evaluate Inverse Sine Functions Example 1

  14. Answer: CHECK If arcsin then sin  Evaluate Inverse Sine Functions Example 1

  15. Evaluate Inverse Sine Functions C. Find the exact value of sin–1 (–2π), if it exists. Because the domain of the inverse sine function is [–1, 1] and –2π < –1, there is no angle with a sine of –2π. Therefore, the value of sin–1(–2π) does not exist. Answer: does not exist Example 1

  16. A. 0 B. C. D. π Find the exact value of sin–1 0. Example 1

  17. Evaluate Inverse Cosine Functions A. Find the exact value of cos–11, if it exists. Find a point on the unit circle on the interval [0, π] with an x-coordinate of 1. When t = 0, cos t = 1. Therefore, cos–11 = 0. Example 2

  18. Evaluate Inverse Cosine Functions Answer:0 Check If cos–1 1 = 0, then cos 0 = 1.  Example 2

  19. B. Find the exact value of , if it exists. Find a point on the unit circle on the interval [0, π] with an x-coordinate of When t = Therefore, arccos Evaluate Inverse Cosine Functions Example 2

  20. Answer: CHECKIf arcos  Evaluate Inverse Cosine Functions Example 2

  21. Evaluate Inverse Cosine Functions C. Find the exact value of cos–1(–2), if it exists. Since the domain of the inverse cosine function is [–1, 1] and –2 < –1, there is no angle with a cosine of –2. Therefore, the value of cos–1(–2) does not exist. Answer:does not exist Example 2

  22. A. B. C. π D. Find the exact value of cos–1 (–1). Example 2

  23. A. Find the exact value of , if it exists. Find a point on the unit circle on the interval such that When t = , tan t = Therefore, Evaluate Inverse Tangent Functions Example 3

  24. Answer: Check If , then tan  Evaluate Inverse Tangent Functions Example 3

  25. Find a point on the unit circle in the interval such that When t = , tan t = Therefore, arctan 1 = . Evaluate Inverse Tangent Functions B. Find the exact value of arctan 1, if it exists. Example 3

  26. Answer: Check If arctan 1 = , then tan = 1.  Evaluate Inverse Tangent Functions Example 3

  27. Find the exact value of arctan . A. B. C. D. Example 3

  28. Key Concept 4

  29. Sketch the graph of y = arctan By definition, y = arctan and tan y = are equivalent on for < y < , so their graphs are the same. Rewrite tan y = as x = 2 tan y and assign values to y on the interval to make a table to values. Sketch Graphs of Inverse Trigonometric Functions Example 4

  30. Sketch Graphs of Inverse Trigonometric Functions Then plot the points (x, y) and connect them with a smooth curve. Notice that this curve is contained within its asymptotes. Answer: Example 4

  31. A. B. C. D. Sketch the graph of y = sin–1 2x. Example 4

  32. Use an Inverse Trigonometric Function A. MOVIES In a movie theater, a 32-foot-tall screen is located 8 feet above ground. Write a function modeling the viewing angle θ for a person in the theater whose eye-level when sitting is 6 feet above ground. Draw a diagram to find the measure of the viewing angle. Let θ1 represent the angle formed from eye-level to the bottom of the screen, and let θ2 represent the angle formed from eye-level to the top of the screen. Example 5

  33. Use an Inverse Trigonometric Function So, the viewing angle is θ = θ2 – θ1. You can use the tangent function to find θ1 and θ2. Because the eye-level of a seated person is 6 feet above the ground, the distance opposite θ1 is 8 – 6 feet or 2 feet long. Example 5

  34. opp = 2 and adj = d Inverse tangent function opp = 34 and adj = d Inverse tangent function Use an Inverse Trigonometric Function The distance opposite θ2 is (32 + 8) – 6 feet or 34 feet Example 5

  35. So, the viewing angle can be modeled by Answer: Use an Inverse Trigonometric Function Example 5

  36. Use an Inverse Trigonometric Function B. MOVIES In a movie theater, a 32-foot-tall screen is located 8 feet above ground-level. Determine the distance that corresponds to the maximum viewing angle. The distance at which the maximum viewing angle occurs is the maximum point on the graph. You can use a graphing calculator to find this point. Example 5

  37. Use an Inverse Trigonometric Function From the graph, you can see that the maximum viewing angle occurs approximately 8.2 feet from the screen. Answer:about 8.2 ft Example 5

  38. A. B. C. D. MATH COMPETITION In a classroom, a 4 foot tall screen is located 6 feet above the floor. Write a function modeling the viewing angle θ for a student in the classroom whose eye-level when sitting is 3 feet above the floor. Example 5

  39. Key Concept 6

  40. A. Find the exact value of , if it exists. The inverse property applies because lies on the interval [–1, 1]. Therefore, Answer: Use Inverse Trigonometric Properties Example 6

  41. B. Find the exact value of , if it exists. Notice that does not lie on the interval [0, π]. However, is coterminal with – 2π or which is on the interval [0, π]. Use Inverse Trigonometric Properties Example 6

  42. Therefore, . Answer: Use Inverse Trigonometric Properties Example 6

  43. C. Find the exact value of , if it exists. Because tan x is not defined when x = , arctan does not exist. Use Inverse Trigonometric Properties Answer:does not exist Example 6

  44. Find the exact value of arcsin A. B. C. D. Example 6

  45. Find the exact value of To simplify the expression, let u = cos–1 so cos u = . Evaluate Compositions of Trigonometric Functions Because the cosine function is positive in Quadrants I and IV, and the domain of the inverse cosine function is restricted to Quadrants I and II, u must lie in Quadrant I. Example 7

  46. Sine function opp = 3 and hyp = 5 So, Answer: Evaluate Compositions of Trigonometric Functions Using the Pythagorean Theorem, you can find that the length of the side opposite  is 3. Now, solve for sin u. Example 7

  47. Find the exact value of A. B. C. D. Example 7

  48. Evaluate Compositions of Trigonometric Functions Write cot (arccos x) as an algebraic expression of x that does not involve trigonometric functions. Let u = arcos x, so cos u = x. Because the domain of the inverse cosine function is restricted to Quadrants I and II, u must lie in Quadrant I or II. The solution is similar for each quadrant, so we will solve for Quadrant I. Example 8

  49. From the Pythagorean Theorem, you can find that the length of the side opposite to u is . Now, solve for cot u. Cotangent function opp = and adj = x So, cot(arcos x) = . Evaluate Compositions of Trigonometric Functions Example 8

  50. Answer: Evaluate Compositions of Trigonometric Functions Example 8

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