408 CHAPTER 6 Trigonometric Functions ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 6.2 Assess Your Understanding 1. In a right triangle, with legs a and b and hypotenuse c, the Pythagorean Theorem states that . (p. A14) 2. The value of the function f x x3 7 ( ) = − at 5 is . (pp. 65–67) 3. True or False For a function y f x , ( ) = for each x in the domain, there is exactly one element y in the range. (pp. 65–66) 4. If two triangles are similar, then corresponding angles are and the lengths of corresponding sides are . (pp. A16–A19) 5. What point is symmetric with respect to the y -axis to the point 1 2 , 3 2 ? ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ (pp. 21–22) 6. If x y , ( ) is a point on the unit circle in quadrant IV and if x 3 2 , = what is y ? (p. 49) Concepts and Vocabulary 7. Multiple Choice Which function takes as input a real number t that corresponds to a point P x y , ( ) = on the unit circle and outputs the x -coordinate? (a) sine (b) cosine (c) tangent (d) secant 8. The point on the unit circle that corresponds to 2 θ π = is P = . 9. The point on the unit circle that corresponds to 4 θ π = is P = . 10. True or False Exact values can be found for the sine of any angle. 11. For any angle θ in standard position, let P x y , ( ) = be the point on the terminal side of θ that is also on the circle x y r . 2 2 2 + = Then, sinθ = and cosθ = . 12. Multiple Choice The point on the unit circle that corresponds to 3 θ π = is (a) 1 2 , 3 2 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ (b) 2 2 , 2 2 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ (c) 3 2 , 1 2 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ (d) 3, 2 3 3 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ Skill Building In Problems 13–20, P x y , ( ) = is the point on the unit circle that corresponds to a real number t. Find the exact values of the six trigonometric functions of t. 13. 3 2 , 1 2 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 14. 1 2 , 3 2 − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 15. 2 5 , 21 5 − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 16. 1 5 , 2 6 5 − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 17. 2 2 , 2 2 − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 18. 2 2 , 2 2 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 19. 2 2 3 , 1 3 − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 20. 5 3 , 2 3 − − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ In Problems 21–30, find the exact value. Do not use a calculator. 21. sin 11 2 π 22. cos 7π ( ) 23. tan 6π ( ) 24. cot 7 2 π 25. csc 11 2 π 26. sec 8π ( ) 27. cos 3 2 π ( ) − 28. sin 3π ( ) − 29. sec π ( ) − 30. tan 3π ( ) − In Problems 31–46, find the exact value of each expression. Do not use a calculator. 31. sin45 cos60 ° + ° 32. sin30 cos45 ° − ° 33. sin90 tan45 ° + ° 34. cos180 sin180 ° − ° 35. sin45 cos45 ° ° 36. tan45 cos30 ° ° 37. csc45 tan60 ° ° 38. sec30 cot45 ° ° 39. 4 sin90 3 tan180 ° − ° 40. 5 cos90 8 sin270 ° − ° 41. 2 sin 3 3 tan 6 π π − 42. π π + 2 sin 4 3 tan 4 43. 2 sec 4 4 cot 3 π π + 44. 3 csc 3 cot 4 π π + 45. csc 2 cot 2 π π + 46. sec csc 2 π π − In Problems 47–64, find the exact values of the six trigonometric functions of the given angle. If any are not defined, say “not defined.” Do not use a calculator. 47. 2 3 π 48. 5 6 π 49. 210° 50. 240° 51. 3 4 π 52. 11 4 π 53. 8 3 π 54. 13 6 π 55. 405° 56. 390° 57. 6 π − 58. 3 π − 59. 135 − ° 60. 240 − ° 61. 5 2 π 62. 5π 63. 14 3 π − 64. 13 6 π − 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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