SECTION 7.7 Product-to-Sum and Sum-to-Product Formulas 537 Solution (a) Use formula (7) to get sin 5 sin 3 2 sin 5 3 2 cos 5 3 2 2sin cos4 θ θ θ θ θ θ θ θ ( ) ( ) ( ) − = − + = (b) cos 3 cos 2 2cos 3 2 2 cos 3 2 2 2cos 5 2 cos 2 θ θ θ θ θ θ θ θ ( ) ( ) + = + − = Formula (8) Skill Building 7.7 Assess Your Understanding In Problems 1–6, find the exact value of each expression. 1. sin195 cos75 ° ⋅ ° 2. cos285 cos195 ° ⋅ ° 3. sin285 sin 75 ° ⋅ ° 4. sin 75 sin15 ° + ° 5. ° − ° cos255 cos195 6. sin255 sin15 ° − ° In Problems 7–16, express each product as a sum containing only sines or only cosines. 7. sin4 sin2 θ θ ( ) ( ) 8. cos4 cos2 θ θ ( ) ( ) 9. sin4 cos2 θ θ ( ) ( ) 10. sin3 sin5 θ θ ( ) ( ) 11. cos3 cos5 θ θ ( ) ( ) 12. sin4 cos6 θ θ ( ) ( ) 13. sin sin2 θ θ ( ) 14. cos3 cos4 θ θ ( ) ( ) 15. sin 3 2 cos 2 θ θ 16. sin 2 cos 5 2 θ θ In Problems 17–24, express each sum or difference as a product of sines and/or cosines. 17. sin 4 sin 2 θ θ ( ) ( ) − 18. sin 4 sin 2 θ θ ( ) ( ) + 19. cos 2 cos 4 θ θ ( ) ( ) + 20. cos 5 cos 3 θ θ ( ) ( ) − 21. sin sin 3 θ θ ( ) + 22. cos cos 3 θ θ ( ) + 23. cos 2 cos 3 2 θ θ − 24. sin 2 sin 3 2 θ θ − In Problems 25–42, establish each identity. 25. sin sin 3 2 sin 2 cos θ θ θ θ ( ) ( ) + = 26. cos cos 3 2cos 2 cos θ θ θ θ ( ) ( ) + = 27. sin 4 sin 2 cos 4 cos 2 tan 3 θ θ θ θ θ ( ) ( ) ( ) ( ) ( ) + + = 28. cos cos 3 sin 3 sin tan 2 θ θ θ θ θ ( ) ( ) ( ) − − = 29. cos cos 3 sin sin 3 tan θ θ θ θ θ ( ) ( ) − + = 30. cos cos 5 sin sin 5 tan 2 θ θ θ θ θ ( ) ( ) ( ) − + = 31. sin sin sin 3 cos cos cos 3 θ θ θ θ θ θ ( )] [ ( ) [ ] + = − 32. sin sin 3 sin 5 cos cos 3 cos 5 θ θ θ θ θ θ ( ) ( )] [ ( ) ( ) [ ] + = − 33. sin 4 sin 8 cos 4 cos 8 tan 6 θ θ θ θ θ ( ) ( ) ( ) ( ) ( ) + + = 34. sin 4 sin 8 cos 4 cos 8 cot 6 θ θ θ θ θ ( ) ( ) ( ) ( ) ( ) − − = − 35. sin 4 sin 8 sin 4 sin 8 tan 6 tan 2 θ θ θ θ θ θ ( ) ( ) ( ) ( ) ( ) ( ) + − = − 36. cos 4 cos 8 cos 4 cos 8 tan2 tan6 θ θ θ θ θ θ ( ) ( ) ( ) ( ) ( ) ( ) − + = 37. sin sin sin sin tan 2 cot 2 α β α β α β α β + − = + − 38. cos cos cos cos cot 2 cot 2 α β α β α β α β + − = − + − 39. sin sin cos cos tan 2 α β α β α β + + = + 40. sin sin cos cos cot 2 α β α β α β − − = − + 41. 1 cos2 cos4 cos6 4 cos cos 2 cos 3 θ θ θ θ θ θ ( ) ( ) ( ) ( ) ( ) + + + = 42. 1 cos2 cos4 cos6 4sin cos2 sin3 θ θ θ θ θ θ ( ) ( ) ( ) ( ) ( ) − + − = 43. Show that sin cos 1 16 1 32 cos 2 1 16 cos 4 1 32 cos 6 . 4 2 θ θ θ θ θ ( ) ( ) ( ) = − − + 44. Show that sin cos 1 16 1 32 cos 2 1 16 cos 4 1 32 cos 6 . 2 4 θ θ θ θ θ ( ) ( ) ( ) = + − − 45. Show that sin 5 16 15 32 cos 2 3 16 cos 4 1 32 cos 6 . 6 θ θ θ θ ( ) ( ) ( ) = − + − 46. Show that cos 5 16 15 32 cos 2 3 16 cos 4 1 32 cos 6 . 6 θ θ θ θ ( ) ( ) ( ) = + + + Now Work PROBLEM 17 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
RkJQdWJsaXNoZXIy NjM5ODQ=