Hprec9 3

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Hprec9 3

  1. 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. 2. Analysis Does sin 90º =2 sin 45°? Does cos 30°= cos 60°? 1 2
  3. 3. Analysis sin( ) sin cos cos sin α β α β α β + = + From Lesson 9-2: Let , then…= = xα β
  4. 4. Important Idea sin 2 2sin cos=x x x Double Angle Identity for sine:
  5. 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. 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. 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. 8. Analysis cos( ) cos cos sin sin α β α β α β + = − From Lesson 9-2: Let , then…= = xα β
  9. 9. Important Idea 2 2 cos2 cos sin= −x x x Double Angle Identity for cosine:
  10. 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. 11. Example If sin x is 1/5 and x is in the first quadrant, find the exact value of cos 2x.
  12. 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. 13. What would you do to derive a double angle identity for tangent?
  14. 14. Double Angle Identity for Tangent: 2 2tan tan 2 1 tan = − x x x Important Idea
  15. 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. 16. Analysis 2 cos2 2cos 1θ θ= − 1 cos2 cos 2 θ θ + = ± One of the double angle identities: Solve for cos θ :
  17. 17. Analysis 1 cos2 os 2 c θ θ + = ± Let 2 2 = ⇒ = x x θ θ
  18. 18. Analysis 1 cos cos 2 2 + = ± x x which results in: The half angle identity for cosine.
  19. 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. 20. Example Use a half angle identity to find the exact value of sin 22°30’ Hint: Multiply the given angle by 2 then divide by 2
  21. 21. Example Use a half angle identity to find the exact value of 19 cos 12 π Hint: Multiply the given angle by 2 then divide by 2
  22. 22. Try This Use a half angle identity to find the exact value of cos 105° 210 2 3 cos105 cos 2 2 ° − ° = = −
  23. 23. Try This Use a half angle identity to find the exact value of 5 5 2 24sin sin 8 2 2 π π + = = 5 sin 8 π

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