Trigonometry addition & substraction id

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Trigonometry addition & substraction id

  1. 1. ADDITION ANDSUBSTRACTION IDENTITIES
  2. 2. Trigonometric Addition Identities For Sine AndCosine
  3. 3. Proof of theyCA O BabcPxα βyCO BacβPxPbCOAα
  4. 4. example
  5. 5. Proof of theexample
  6. 6. 90-AA yxzProof of theexample
  7. 7. Proof of the
  8. 8. Quadrant IαQuadrant II1800 - αQuadrant III1800+ αQuadrant IV3600 - αsin + + - -cos + - - +tan + - + -SinTancossintan cos
  9. 9. Double angleidentities
  10. 10. By using the result for sin2αinto our RHS and obtain:(remember:example
  11. 11. Change sin 70◦cos 150 ◦+cos70 ◦sin150◦into atrigonometric function in a single variable andevaluate it.Answer:This is one side of sum idnetity for sines :sin(α+β)=sin α.cos β+cos α.sin βsin 70◦cos 150 ◦+cos70 ◦sin150◦ = sin (70◦+150◦ )= sin (220◦)= -sin 220◦=-sin (220 ◦ - 180 ◦)=-sin 40 ◦=-.643180 ◦ <220 ◦ <270 ◦Quadrant lll=-cos(270-220)=-cos 50=-.643
  12. 12. Change sin 60◦cos 45 ◦- cos 60 ◦sin45◦into atrigonometric function in a single variable andevaluate it.Answer:This is part of the difference identity for sines :sin(α-β)=sin α.cos β-cos α.sin βsin 60◦cos 45 ◦- cos 60 ◦sin150◦ = sin (60◦-45◦ )= sin (15◦)= sin 15 ◦=.259
  13. 13. Change cos 85◦cos 15 ◦- sin85◦ sin15◦into atrigonometric function in a single variable andevaluate it.Answer:This is part of the sum identity for cosines :cos(α+β) = cos α.cos β – sin α.sin βcos 85◦cos 15 ◦- sin85◦ sin15◦ = cos (85◦+15◦ )= cos (100◦)= -cos 100 ◦=-cos(180 ◦ - 100◦)=-cos 80 ◦=-.087
  14. 14. 453αIf P in the second quadrant and sinP= , find sin2P.Answer:If sinP= , then cosP=sin2P = 2sinP.cosPsin2P= 2=
  15. 15. If A is a second quadrant angle, and sinA = what is cos2A ?Answaer:Use pythagorean tripels : (5, 12, 13)If sinA= , then cosA=If we want to use the formula of :Cause cosine in negative inquadrant II→

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