SECTION 7.6 Double-angle and Half-angle Formulas 531 Skill Building In Problems 9–20, use the information given about the angle θ θ π ≤ < , 0 2 , to find the exact value of : (a) θ ( ) sin 2 (b) θ ( ) cos 2 (c) θ sin 2 (d) θ cos 2 (e) θ ( ) tan 2 (f) θ tan 2 9. θ θ π = < < sin 3 5 , 0 2 10. θ θ π = < < cos 3 5 , 0 2 11. θ π θ π = < < tan 4 3 , 3 2 12. θ π θ π = < < tan 1 2 , 3 2 13. θ π θ π = − < < cos 6 3 , 2 14. θ π θ π = − < < sin 3 3 , 3 2 2 15. θ θ = > sec 3, sin 0 16. θ θ = − < csc 5 , cos 0 17. θ θ = − < cot 2, sec 0 18. θ θ = < sec 2, csc 0 19. θ θ = − < tan 3, sin 0 20. θ θ = < cot 3, cos 0 In Problems 21–30, use Half-angle Formulas to find the exact value of each expression. 21. ° sin22.5 22. ° cos22.5 23. π tan 7 8 24. π tan 9 8 25. ° cos165 26. ° sin195 27. π sec 15 8 28. π csc 7 8 29. π ( ) − sin 8 30. π ( ) − cos 3 8 In Problems 31 – 42 , ( ) = f x x sin , ( ) = g x x cos , and ( ) = h x x tan . Use the figures below to evaluate each function. y x (a, 2) x2 1 y2 5 5 u y x (2 , b) x2 1 y2 5 1 a 1 – 4 31. θ ( ) f 2 32. θ ( ) g 2 33. θ( ) g 2 34. θ( ) f 2 35. θ ( ) h 2 36. θ( ) h 2 37. α ( ) g 2 38. α ( ) f 2 39. α( ) f 2 40. α( ) g 2 41. α( ) h 2 42. α ( ) h 2 43. Show that θ θ θ ( ) ( ) = − + sin 3 8 1 2 cos 2 1 8 cos 4 . 4 44. Show that θ θ θ θ ( ) ( ) ( ) = − sin 4 cos 4 sin 8 sin . 3 45. Show that θ θ θ ( ) = − sin cos 1 8 1 8 cos 4 2 2 . 46. Show that θ θ θ θ ( ) ( ) = − + sin cos 3 128 1 32 cos 4 1 128 cos 8 . 4 4 7. Multiple Choice Choose the expression that completes the Half-angle Formula for cosine functions: α = cos 2 . (a) α ± −1 cos 2 (b) α ± +1 cos 2 (c) α α ± − cos sin 2 (d) α α ± − + 1 cos 1 cos 8. Multiple Choice If sin α θ = ± −1 cos 2 , then which statement describes how θ is related to α? (a) θ α = (b) θ α = 2 (c) θ α = 2 (d) θ α = 2 1. θ θ ( ) = − cos 2 cos2 = −1 = −1 2. sin2 θ = −1 cos 2 3. θ θ = − tan 2 1 cos 4. True or False θ θ θ ( ) = − tan 2 2 tan 1 tan2 5. True or False θ ( ) sin 2 has two equivalent forms: θ θ θ θ − 2 sin cos and sin cos 2 2 6. True or False θ θ θ ( ) ( ) ( ) + = tan 2 tan 2 tan 4 Concepts and Vocabulary 7.6 Assess Your Understanding 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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