0704 ch 7 day 4

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  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • 1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
  • \n
  • 0704 ch 7 day 4

    1. 1. 7.2 Addition & Subtraction FormulasRevelation 22:17The Spirit and the bride say, "Come!" And let him whohears say, "Come!" Whoever is thirsty, let him come;and whoever wishes, let him take the free gift of thewater of life.
    2. 2. Addition & Subtraction FormulasThese are on your help sheet
    3. 3. Addition & Subtraction FormulasThese are on your help sheetTake a look at these ... but why do we have them?
    4. 4. Addition & Subtraction FormulasThese are on your help sheetTake a look at these ... but why do we have them?The Unit Circle allows us to find the trig values for“pretty points” ... like sin(30°) or cos(135°).To get exact trig values for non-pretty angles, wecan use the Addition & Subtraction identities.
    5. 5. Addition & Subtraction FormulasThese are on your help sheetTake a look at these ... but why do we have them?The Unit Circle allows us to find the trig values for“pretty points” ... like sin(30°) or cos(135°).To get exact trig values for non-pretty angles, wecan use the Addition & Subtraction identities.Examples: sin ( 75° ) is sin ( 45° + 30° ) cos ( 265° ) is cos ( 310° − 45° )
    6. 6. Find the exact value of sin(105°)
    7. 7. Find the exact value of sin(105°)We need to find 2 “magic point” angles on the UnitCircle that either have a sum or a difference toequal 105 degrees.
    8. 8. Find the exact value of sin(105°)We need to find 2 “magic point” angles on the UnitCircle that either have a sum or a difference toequal 105 degrees. 105° = 60° + 45°
    9. 9. Find the exact value of sin(105°)We need to find 2 “magic point” angles on the UnitCircle that either have a sum or a difference toequal 105 degrees. 105° = 60° + 45° sin (105° ) = sin ( 60° + 45° )
    10. 10. Find the exact value of sin(105°)We need to find 2 “magic point” angles on the UnitCircle that either have a sum or a difference toequal 105 degrees. 105° = 60° + 45° sin (105° ) = sin ( 60° + 45° ) = sin 60°cos 45° + cos 60°sin 45°
    11. 11. Find the exact value of sin(105°)We need to find 2 “magic point” angles on the UnitCircle that either have a sum or a difference toequal 105 degrees. 105° = 60° + 45° sin (105° ) = sin ( 60° + 45° ) = sin 60°cos 45° + cos 60°sin 45° 3 2 1 2 = ⋅ + ⋅ 2 2 2 2
    12. 12. Find the exact value of sin(105°)We need to find 2 “magic point” angles on the UnitCircle that either have a sum or a difference toequal 105 degrees. 105° = 60° + 45° sin (105° ) = sin ( 60° + 45° ) = sin 60°cos 45° + cos 60°sin 45° 3 2 1 2 = ⋅ + ⋅ 2 2 2 2 6+ 2 = 4
    13. 13. ⎛ 5π ⎞Find the exact value of cos ⎜ ⎟ ⎝ 12 ⎠
    14. 14. ⎛ 5π ⎞Find the exact value of cos ⎜ ⎟ ⎝ 12 ⎠ 8π 3π − 12 12 2π π − 3 4
    15. 15. ⎛ 5π ⎞Find the exact value of cos ⎜ ⎟ ⎝ 12 ⎠ 8π 3π − 12 12 2π π or − 3 4
    16. 16. ⎛ 5π ⎞Find the exact value of cos ⎜ ⎟ ⎝ 12 ⎠ 8π 3π 2π 3π − + 12 12 12 12 2π π or π π − + 3 4 6 4
    17. 17. ⎛ 5π ⎞Find the exact value of cos ⎜ ⎟ ⎝ 12 ⎠ 8π 3π 2π 3π − + 12 12 12 12 2π π or π π − + 3 4 6 4 ⎛ 5π ⎞ ⎛ 2π π ⎞cos ⎜ ⎟ = cos ⎜ − ⎟ ⎝ 12 ⎠ ⎝ 3 4 ⎠
    18. 18. ⎛ 5π ⎞Find the exact value of cos ⎜ ⎟ ⎝ 12 ⎠ 8π 3π 2π 3π − + 12 12 12 12 2π π or π π − + 3 4 6 4 ⎛ 5π ⎞ ⎛ 2π π ⎞cos ⎜ ⎟ = cos ⎜ − ⎟ ⎝ 12 ⎠ ⎝ 3 4 ⎠ 2π π 2π π= cos cos + sin sin 3 4 3 4
    19. 19. ⎛ 5π ⎞Find the exact value of cos ⎜ ⎟ ⎝ 12 ⎠ 8π 3π 2π 3π − + 12 12 12 12 2π π or π π − + 3 4 6 4 ⎛ 5π ⎞ ⎛ 2π π ⎞cos ⎜ ⎟ = cos ⎜ − ⎟ ⎝ 12 ⎠ ⎝ 3 4 ⎠ 2π π 2π π= cos cos + sin sin 3 4 3 4 ⎛ 1 ⎞ ⎛ 2 ⎞ ⎛ 3 ⎞ ⎛ 2 ⎞= ⎜ − ⎟ ⎜ ⎟ + ⎜ 2 ⎟ ⎜ 2 ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ ⎠ ⎝ ⎠
    20. 20. ⎛ 5π ⎞Find the exact value of cos ⎜ ⎟ ⎝ 12 ⎠ 8π 3π 2π 3π − + 12 12 12 12 2π π or π π − + 3 4 6 4 ⎛ 5π ⎞ ⎛ 2π π ⎞cos ⎜ ⎟ = cos ⎜ − ⎟ ⎝ 12 ⎠ ⎝ 3 4 ⎠ 2 6 =− + 4 4 2π π 2π π= cos cos + sin sin 3 4 3 4 ⎛ 1 ⎞ ⎛ 2 ⎞ ⎛ 3 ⎞ ⎛ 2 ⎞= ⎜ − ⎟ ⎜ ⎟ + ⎜ 2 ⎟ ⎜ 2 ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ ⎠ ⎝ ⎠
    21. 21. ⎛ 5π ⎞Find the exact value of cos ⎜ ⎟ ⎝ 12 ⎠ 8π 3π 2π 3π − + 12 12 12 12 2π π or π π − + 3 4 6 4 ⎛ 5π ⎞ ⎛ 2π π ⎞cos ⎜ ⎟ = cos ⎜ − ⎟ ⎝ 12 ⎠ ⎝ 3 4 ⎠ 2 6 =− + 4 4 2π π 2π π= cos cos + sin sin 3 4 3 4 6− 2 = ⎛ 1 ⎞ ⎛ 2 ⎞ ⎛ 3 ⎞ ⎛ 2 ⎞ 4= ⎜ − ⎟ ⎜ ⎟ + ⎜ 2 ⎟ ⎜ 2 ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ ⎠ ⎝ ⎠
    22. 22. Prove the identity:
    23. 23. Prove the identity: ⎛ π ⎞ 2 ( cos α + sin α ) cos ⎜ α − ⎟ = ⎝ 4 ⎠ 2
    24. 24. Prove the identity: ⎛ π ⎞ 2 ( cos α + sin α ) cos ⎜ α − ⎟ = ⎝ 4 ⎠ 2 π πcos α cos + sin α sin = 4 4
    25. 25. Prove the identity: ⎛ π ⎞ 2 ( cos α + sin α ) cos ⎜ α − ⎟ = ⎝ 4 ⎠ 2 π πcos α cos + sin α sin = 4 4 2 2 cos α + sin α = 2 2
    26. 26. Prove the identity: ⎛ π ⎞ 2 ( cos α + sin α ) cos ⎜ α − ⎟ = ⎝ 4 ⎠ 2 π πcos α cos + sin α sin = 4 4 2 2 cos α + sin α = 2 2 2 ( cos α + sin α ) = 2
    27. 27. Prove the identity:
    28. 28. Prove the identity: ⎛ π ⎞ tan θ + 1 tan ⎜ θ + ⎟ = ⎝ 4 ⎠ 1− tan θ
    29. 29. Prove the identity: ⎛ π ⎞ tan θ + 1 tan ⎜ θ + ⎟ = ⎝ 4 ⎠ 1− tan θ π tan θ + tan 4 = π 1− tan θ tan 4
    30. 30. Prove the identity: ⎛ π ⎞ tan θ + 1 tan ⎜ θ + ⎟ = ⎝ 4 ⎠ 1− tan θ π tan θ + tan 4 = π 1− tan θ tan 4 tan θ + 1 = 1− tan θ (1)
    31. 31. Prove the identity: ⎛ π ⎞ tan θ + 1 tan ⎜ θ + ⎟ = ⎝ 4 ⎠ 1− tan θ π tan θ + tan 4 = π 1− tan θ tan 4 tan θ + 1 = 1− tan θ (1) tan θ + 1 = 1− tan θ
    32. 32. We are skipping the part entitled, Expressions of the Form Asinx + Bcosx
    33. 33. We are skipping the part entitled, Expressions of the Form Asinx + BcosxIf time allows, let’s derive the Difference Identity for Cosine
    34. 34. We are skipping the part entitled, Expressions of the Form Asinx + BcosxIf time allows, let’s derive the Difference Identity for CosineOur procedure will be:
    35. 35. We are skipping the part entitled, Expressions of the Form Asinx + BcosxIf time allows, let’s derive the Difference Identity for CosineOur procedure will be: 1. use law of cosines to find AB
    36. 36. We are skipping the part entitled, Expressions of the Form Asinx + BcosxIf time allows, let’s derive the Difference Identity for CosineOur procedure will be: 1. use law of cosines to find AB 2. use distance formula to find AB
    37. 37. We are skipping the part entitled, Expressions of the Form Asinx + BcosxIf time allows, let’s derive the Difference Identity for CosineOur procedure will be: 1. use law of cosines to find AB 2. use distance formula to find AB 3. equate these two results and solve
    38. 38. cos (α − β ) = cos α cos β + sin α sin β
    39. 39. cos (α − β ) = cos α cos β + sin α sin β1. Use Law of Cosines to find AB
    40. 40. cos (α − β ) = cos α cos β + sin α sin β1. Use Law of Cosines to find AB 2 2 2 (AB) = 1 + 1 − 2(1)(1)cos (α − β )
    41. 41. cos (α − β ) = cos α cos β + sin α sin β1. Use Law of Cosines to find AB 2 2 2 (AB) = 1 + 1 − 2(1)(1)cos (α − β ) 2 (AB) = 2 − 2 cos (α − β )
    42. 42. cos (α − β ) = cos α cos β + sin α sin β1. Use Law of Cosines to find AB 2 2 2 (AB) = 1 + 1 − 2(1)(1)cos (α − β ) 2 (AB) = 2 − 2 cos (α − β ) AB = 2 − 2 cos (α − β )
    43. 43. 2. Use Distance Formula to find AB
    44. 44. 2. Use Distance Formula to find AB 2 2 AB = ( cos α − cos β ) + (sin α − sin β )
    45. 45. 2. Use Distance Formula to find AB 2 2 AB = ( cos α − cos β ) + (sin α − sin β ) 2 2 2 2AB = cos α − 2 cos α cos β + cos β + sin α − 2sin α sin β + sin β
    46. 46. 2. Use Distance Formula to find AB 2 2 AB = ( cos α − cos β ) + (sin α − sin β ) 2 2 2 2AB = cos α − 2 cos α cos β + cos β + sin α − 2sin α sin β + sin βAB = (sin α + cos α ) + (sin 2 2 2 β + cos β ) − 2 cos α cos β − 2sin α sin β 2
    47. 47. 2. Use Distance Formula to find AB 2 2 AB = ( cos α − cos β ) + (sin α − sin β ) 2 2 2 2AB = cos α − 2 cos α cos β + cos β + sin α − 2sin α sin β + sin βAB = (sin α + cos α ) + (sin 2 2 2 β + cos β ) − 2 cos α cos β − 2sin α sin β 2 AB = 2 − 2 cos α cos β − 2sin α sin β
    48. 48. 3. Equate the two and solve
    49. 49. 3. Equate the two and solve 2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β
    50. 50. 3. Equate the two and solve 2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β 2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β
    51. 51. 3. Equate the two and solve 2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β 2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β −2 cos (α − β ) = −2 cos α cos β − 2sin α sin β
    52. 52. 3. Equate the two and solve 2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β 2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β −2 cos (α − β ) = −2 cos α cos β − 2sin α sin β −2 cos (α − β ) = −2 ( cos α cos β + sin α sin β )
    53. 53. 3. Equate the two and solve 2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β 2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β −2 cos (α − β ) = −2 cos α cos β − 2sin α sin β −2 cos (α − β ) = −2 ( cos α cos β + sin α sin β ) cos (α − β ) = cos α cos β + sin α sin β
    54. 54. 3. Equate the two and solve 2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β 2 − 2 cos (α − β ) = 2 − 2 cos α cos β − 2sin α sin β −2 cos (α − β ) = −2 cos α cos β − 2sin α sin β −2 cos (α − β ) = −2 ( cos α cos β + sin α sin β ) cos (α − β ) = cos α cos β + sin α sin β And this is the Difference Identity for CosineWe could derive all the others in a similar fashion ...
    55. 55. HW #4Today is just a good day in disguise. Paul Venghaus

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