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Transcript of Hprec9 3

• 1. 9-3: Other Identities Objectives: Use the double angle and half angle identities for sine, cosine & tangent 2009 Roy L. Gover (www.mrgover.com) )

2. Analysis Does sin 90 =2 sin 45? Does cos 30= cos 60? 1 2 3. Analysis sin( ) sin cos cos sin + = + From Lesson 9-2: Let , then= = x 4. Important Idea sin 2 2sin cos=x x x Double Angle Identity for sine: 5. Example If sin x = 4/5 and x is in the first quadrant, find the exact value of sin 2x. first, find cos x 6. Try This If sin x = 3/5 and x is in the first quadrant, find the exact value of sin 2x 24 sin 2 25 =x 7. Try This If sin x = 3/5 and x is in the second quadrant, find the exact value of sin 2x. 24 sin 2 25 = x 8. Analysis cos( ) cos cos sin sin + = From Lesson 9-2: Let , then= = x 9. Important Idea 2 2 cos2 cos sin= x x x Double Angle Identity for cosine: 10. Double Angle Identity for cos Three forms... 1. cos 2x = cos2 x - sin2 x 2. cos 2x = 2cos2 x - 1 3. cos 2x = 1 - 2sin2 x Important Idea 11. Example If sin x is 1/5 and x is in the first quadrant, find the exact value of cos 2x. 12. Try This If sin x is 2/5 and x is in the second quadrant, find the exact value of cos 2x. 17 cos2 25 =x 13. What would you do to derive a double angle identity for tangent? 14. Double Angle Identity for Tangent: 2 2tan tan 2 1 tan = x x x Important Idea 15. Try This If , and is in the first quadrant, find . 5 cos 5 =x x tan 2x Hint: first find tan x 4 tan 2 3 = x 16. Analysis 2 cos2 2cos 1 = 1 cos2 cos 2 + = One of the double angle identities: Solve for cos : 17. Analysis 1 cos2 os 2 c + = Let 2 2 = = x x 18. Analysis 1 cos cos 2 2 + = x x which results in: The half angle identity for cosine. 19. Important Idea 1 cos sin 2 2 = x x 1 cos cos 2 2 + = x x 1 cos tan 2 1 cos = + x x x The half angle identities: See page 596 of your text. 20. Example Use a half angle identity to find the exact value of sin 2230 Hint: Multiply the given angle by 2 then divide by 2 21. Example Use a half angle identity to find the exact value of 19 cos 12 Hint: Multiply the given angle by 2 then divide by 2 22. Try This Use a half angle identity to find the exact value of cos 105 210 2 3 cos105 cos 2 2 = = 23. Try This Use a half angle identity to find the exact value of 5 5 2 24sin sin 8 2 2 + = = 5 sin 8