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Splash Screen. Five-Minute Check (over Lesson 12–8) CCSS Then/Now New Vocabulary Key Concept: Inverse Trigonometric Functions Example 1: Evaluate Inverse Trigonometric Functions Example 2: Find a Trigonometric Value Example 3: Standardized Test Example

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 12–8) CCSS Then/Now New Vocabulary Key Concept: Inverse Trigonometric Functions Example 1: Evaluate Inverse Trigonometric Functions Example 2: Find a Trigonometric Value Example 3: Standardized Test Example Example 4: Real-World Example: Use Inverse Trigonometric Functions Lesson Menu

  3. State the phase shift for y = 2 cos (x + 60°). A. –60° B. –30° C. 60° D. 120° 5-Minute Check 1

  4. State the phase shift for y = 2 cos (x + 60°). A. –60° B. –30° C. 60° D. 120° 5-Minute Check 1

  5. State the vertical shift for y = 2 sin 3x + . A.– B.2 C. D. 1 1 1 1 __ __ __ __ 2 2 2 4 5-Minute Check 2

  6. State the vertical shift for y = 2 sin 3x + . A.– B.2 C. D. 1 1 1 1 __ __ __ __ 2 2 2 4 5-Minute Check 2

  7. State the amplitude of y = –1 + 3 sin A. 1 B. 2 C. 3 D. 4 5-Minute Check 3

  8. State the amplitude of y = –1 + 3 sin A. 1 B. 2 C. 3 D. 4 5-Minute Check 3

  9. State the period of y = cos A. 180º B. 1080º C. 2160º D. 3240º 5-Minute Check 4

  10. State the period of y = cos A. 180º B. 1080º C. 2160º D. 3240º 5-Minute Check 4

  11. Identify the equation of a sine function that has a graph that is the graph of the parent function translated 5 units up and 3 units left. A.y = 5 sin (x + 3) B.y = sin (x – 3) + 5 C.y = sin (x + 3) + 5 D.y = 3 sin (x + 5) 5-Minute Check 5

  12. Identify the equation of a sine function that has a graph that is the graph of the parent function translated 5 units up and 3 units left. A.y = 5 sin (x + 3) B.y = sin (x – 3) + 5 C.y = sin (x + 3) + 5 D.y = 3 sin (x + 5) 5-Minute Check 5

  13. Content Standards A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Mathematical Practices 7 Look for and make use of structure. CCSS

  14. You graphed trigonometric functions. • Find values of inverse trigonometric functions. • Solve equations by using inverse trigonometric functions. Then/Now

  15. principal values • Arcsine function • Arccosine function • Arctangent function Vocabulary

  16. Concept

  17. A.Find the value of . Write the angle measure in degrees and radians. Find the angle  for –90° ≤  ≤ 90° that has a sine value of . Evaluate Inverse Trigonometric Functions Example 1

  18. Find a point on the unit circle that has an x-coordinate of . When  = 45°, sin  = . So, = 45° or . Evaluate Inverse Trigonometric Functions Method 1 Use a unit circle. Example 1

  19. Keystrokes: ) ) 2nd 2nd ÷ [SIN–1] [ ] 2 2 45 ENTER Evaluate Inverse Trigonometric Functions Method 2 Use a calculator. Answer: Example 1

  20. Keystrokes: ) ) 2nd 2nd ÷ [SIN–1] [ ] 2 2 45 ENTER Answer: Therefore, = 45° or . Evaluate Inverse Trigonometric Functions Method 2 Use a calculator. Example 1

  21. ) 2nd (–) ENTER Keystrokes: [SIN–1] 1 –90 Evaluate Inverse Trigonometric Functions B.Find the value of Arcsin (–1). Write the angle measure in degrees and radians. Find the angle  for –90° ≤  ≤ 90° that has a sine value of –1. Answer: Example 1

  22. ) 2nd (–) ENTER Keystrokes: [SIN–1] 1 –90 Answer: Therefore, Arcsin (–1) = –90° or – . Evaluate Inverse Trigonometric Functions B.Find the value of Arcsin (–1). Write the angle measure in degrees and radians. Find the angle  for –90° ≤  ≤ 90° that has a sine value of –1. Example 1

  23. A. Find the value of Cos–1 . Write the angle measure in degrees and radians. A.60° or B.90° or C.180° or  D.45° or Example 1

  24. A. Find the value of Cos–1 . Write the angle measure in degrees and radians. A.60° or B.90° or C.180° or  D.45° or Example 1

  25. A.60° or B.90° or C.180° or  D.45° or B. Find the value of Arccos(–1). Write the angle measure in degrees and radians. Example 1

  26. A.60° or B.90° or C.180° or  D.45° or B. Find the value of Arccos(–1). Write the angle measure in degrees and radians. Example 1

  27. Find the value of Round to the nearest hundredth. Keystrokes: 7 4 ) TAN 2nd ) [COS–1] ÷ ENTER 1.436140662 Find a Trigonometric Value Answer: Example 2

  28. Find the value of Round to the nearest hundredth. Keystrokes: 7 4 ) TAN 2nd ) [COS–1] ÷ ENTER 1.436140662 Find a Trigonometric Value Answer: 1.44 Example 2

  29. Find the value of Write the angle measure in radians. Round to the nearest hundredth. A. 1.5 radians B. 1.12 radians C. 1.04 radians D. 1.62 radians Example 2

  30. Find the value of Write the angle measure in radians. Round to the nearest hundredth. A. 1.5 radians B. 1.12 radians C. 1.04 radians D. 1.62 radians Example 2

  31. If cos  = –0.86, find . A –149.3° B –59.3° C 59.3° D 149.3° Read the Test Item The cosine of angle  is 0.86. This can be written as Arccos (0.86) = . Example 3

  32. Keystrokes: 2nd ENTER 149.3165829 ) [COS–1] (–) 0.86 Solve the Test Item Use a calculator. Answer: Example 3

  33. Keystrokes: 2nd ENTER 149.3165829 ) [COS–1] (–) 0.86 Solve the Test Item Use a calculator. Answer: So,  = 149.3. The answer is D. Example 3

  34. If sin  = –0.707, find . A. –30° B. –45° C. 60° D. 75° Example 3

  35. If sin  = –0.707, find . A. –30° B. –45° C. 60° D. 75° Example 3

  36. Use Inverse Trigonometric Functions WATER SKIING A water ski ramp is 5.5 feet tall and 12 feet long, as shown below. Write an inverse trigonometric function that can be used to find . Then find the measure of . Round to the nearest tenth. Example 4

  37. Use Inverse Trigonometric Functions Because the measures of the opposite side and the hypotenuse are known, you can use the sine function. Sine function Inverse sine function  ≈ 27.3° Use a calculator. Answer: Example 4

  38. Use Inverse Trigonometric Functions Because the measures of the opposite side and the hypotenuse are known, you can use the sine function. Sine function Inverse sine function  ≈ 27.3° Use a calculator. Answer: So, the angle of the ramp is about 27.3°. Example 4

  39. A. B. C. D. WATER SKIING A water ski ramp is 8 feet tall and 20 feet long. Write an inverse trigonometric function that can be used to find . Then find the measure of . Round to the nearest tenth. Example 4

  40. A. B. C. D. WATER SKIING A water ski ramp is 8 feet tall and 20 feet long. Write an inverse trigonometric function that can be used to find . Then find the measure of . Round to the nearest tenth. Example 4

  41. End of the Lesson

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