Inverse trigonometric functions

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Inverse trigonometric functions

  1. 1. Inverse Trigonometric Functions Mathematics 4 October 24, 20111 of 26
  2. 2. Inverse Trigonometric Functions 3 If sin x = 5 , what is x?2 of 26
  3. 3. Inverse Trigonometric Functions 3 If sin x = 5 , what is x? How do we isolate x from the equation above?2 of 26
  4. 4. Inverse Trigonometric FunctionsLet us recall inverses!• f (x) = y = 2x − 1•• 3 of 26
  5. 5. Inverse Trigonometric FunctionsLet us recall inverses!• f (x) = y = 2x − 1• f −1 (x) →• 3 of 26
  6. 6. Inverse Trigonometric FunctionsLet us recall inverses!• f (x) = y = 2x − 1• f −1 (x) → x = 2y − 1 The variables are interchanged.• 3 of 26
  7. 7. Inverse Trigonometric FunctionsLet us recall inverses!• f (x) = y = 2x − 1• f −1 (x) → x = 2y − 1 The variables are interchanged. x+1• f −1 (x) = y = The y-variable is isolated. 2 3 of 26
  8. 8. Inverse Trigonometric FunctionsLet us recall inverses! Given the graph of g(x): 4 of 26
  9. 9. Inverse Trigonometric FunctionsLet us recall inverses! The inverse of g(x) can be flipping the graph along the diagonal: 4 of 26
  10. 10. Inverse Trigonometric FunctionsLet us recall inverses! This is the graph of g −1 (x) 4 of 26
  11. 11. Inverse Trigonometric FunctionsDoes the function f (x) = sin x have an inverse? f (x) = sin x 5 of 26
  12. 12. Inverse Trigonometric FunctionsDoes the function f (x) = sin x have an inverse? f (x) = sin x No! 5 of 26
  13. 13. Inverse Trigonometric FunctionsDoes the function f (x) = sin x have an inverse? f (x) = sin x No! The function f (x) = sin x is NOT one-to-one! 5 of 26
  14. 14. Inverse Trigonometric FunctionsDoes the function f (x) = sin x have an inverse? f (x) = sin x No! The function f (x) = sin x is NOT one-to-one! It does not pass the Horizontal Line Test! 5 of 26
  15. 15. Inverse Trigonometric FunctionsDo any of the six trigonometric functions have inverses? f (x) = sin x f (x) = cos x 6 of 26
  16. 16. Inverse Trigonometric FunctionsDo any of the six trigonometric functions have inverses? f (x) = tan x f (x) = cot x 6 of 26
  17. 17. Inverse Trigonometric FunctionsDo any of the six trigonometric functions have inverses? f (x) = sec x f (x) = csc x 6 of 26
  18. 18. Inverse Trigonometric FunctionsHow can we isolate x in f (x) = sin x if f (x) is not one-to-one? f (x) = sin x 7 of 26
  19. 19. Inverse Trigonometric FunctionsHow can we isolate x in f (x) = sin x if f (x) is not one-to-one? f (x) = Sin x, x ∈ [− π , π ] 2 2 7 of 26
  20. 20. Inverse Trigonometric FunctionsHow can we isolate x in f (x) = sin x if f (x) is not one-to-one? f (x) = Sin x, x ∈ [− π , π ] 2 2 Restrict the domain so that it becomes one-to-one. 7 of 26
  21. 21. Inverse Trigonometric FunctionsThe inverse of f (x) = Sin x, x ∈ [− π , π ] 2 2 f (x) = Sin x, x ∈ [− π , π ] 2 2 8 of 26
  22. 22. Inverse Trigonometric FunctionsThe inverse of f (x) = Sin x, x ∈ [− π , π ] 2 2 Find the graph of the inverse by flipping along the diagonal 8 of 26
  23. 23. Inverse Trigonometric FunctionsThe inverse of f (x) = Sin x, x ∈ [− π , π ] 2 2 f (x) = sin−1 x = Arcsin x = inverse sine of x 8 of 26
  24. 24. The Inverse Sine FunctionProperties of f (x) = sin−1 x: Domain: Range: 9 of 26
  25. 25. The Inverse Sine FunctionProperties of f (x) = sin−1 x: Domain: x ∈ [−1, 1] Range: 9 of 26
  26. 26. The Inverse Sine FunctionProperties of f (x) = sin−1 x: Domain: x ∈ [−1, 1] Range: y ∈ [− π , π ] 2 2 9 of 26
  27. 27. The Inverse Sine FunctionProperties of f (x) = sin−1 x:10 of 26
  28. 28. The Inverse Sine FunctionDetermine the following values:1. sin−1 1 2 =2. Arcsin 1 =3. sin−1 (sin π ) = 44. Arcsin(sin 7π ) = 65. sin−1 (sin 4π ) = 3 11 of 26
  29. 29. The Inverse Sine FunctionDetermine the following values:1. sin−1 1 2 = π 62. Arcsin 1 =3. sin−1 (sin π ) = 44. Arcsin(sin 7π ) = 65. sin−1 (sin 4π ) = 3 11 of 26
  30. 30. The Inverse Sine FunctionDetermine the following values:1. sin−1 1 2 = π 6 π2. Arcsin 1 = 23. sin−1 (sin π ) = 44. Arcsin(sin 7π ) = 65. sin−1 (sin 4π ) = 3 11 of 26
  31. 31. The Inverse Sine FunctionDetermine the following values:1. sin−1 1 2 = π 6 π2. Arcsin 1 = 23. sin−1 (sin π ) = 4 π 44. Arcsin(sin 7π ) = 65. sin−1 (sin 4π ) = 3 11 of 26
  32. 32. The Inverse Sine FunctionDetermine the following values:1. sin−1 1 2 = π 6 π2. Arcsin 1 = 23. sin−1 (sin π ) = 4 π 44. Arcsin(sin 7π ) = − π 6 65. sin−1 (sin 4π ) = 3 11 of 26
  33. 33. The Inverse Sine FunctionDetermine the following values:1. sin−1 1 2 = π 6 π2. Arcsin 1 = 23. sin−1 (sin π ) = 4 π 44. Arcsin(sin 7π ) = − π 6 65. sin−1 (sin 4π ) = − π 3 3 11 of 26
  34. 34. The Inverse Cosine FunctionGraphing the inverse cosine function f (x) = cos x12 of 26
  35. 35. The Inverse Cosine FunctionGraphing the inverse cosine function f (x) = Cos x, x ∈ [0, π]12 of 26
  36. 36. The Inverse Cosine FunctionGraphing the inverse cosine function f (x) = Cos x, x ∈ [0, π] Restrict the domain so that it becomes one-to-one.12 of 26
  37. 37. The Inverse Cosine FunctionThe inverse of f (x) = Cos x, x ∈ [0, π] f (x) = Cos x, x ∈ [0, π]13 of 26
  38. 38. The Inverse Cosine FunctionThe inverse of f (x) = Cos x, x ∈ [0, π] Find the graph of the inverse by flipping along the diagonal13 of 26
  39. 39. The Inverse Cosine FunctionThe inverse of f (x) = Cos x, x ∈ [0, π] f (x) = cos−1 x = Arccos x = inverse cosine of x13 of 26
  40. 40. The Inverse Cosine FunctionProperties of f (x) = cos−1 x: Domain: Range:14 of 26
  41. 41. The Inverse Cosine FunctionProperties of f (x) = cos−1 x: Domain: x ∈ [−1, 1] Range:14 of 26
  42. 42. The Inverse Cosine FunctionProperties of f (x) = cos−1 x: Domain: x ∈ [−1, 1] Range: y ∈ [0, π]14 of 26
  43. 43. The Inverse cosine FunctionProperties of f (x) = cos−1 x:15 of 26
  44. 44. The Inverse Cosine FunctionDetermine the following values:1. cos−1 1 2 =2. Arccos 0 =3. Arccos(cos 7π ) = 64. cos−1 (cos 7π ) = 416 of 26
  45. 45. The Inverse Cosine FunctionDetermine the following values:1. cos−1 1 2 = π 32. Arccos 0 =3. Arccos(cos 7π ) = 64. cos−1 (cos 7π ) = 416 of 26
  46. 46. The Inverse Cosine FunctionDetermine the following values:1. cos−1 1 2 = π 3 π2. Arccos 0 = 23. Arccos(cos 7π ) = 64. cos−1 (cos 7π ) = 416 of 26
  47. 47. The Inverse Cosine FunctionDetermine the following values:1. cos−1 1 2 = π 3 π2. Arccos 0 = 23. Arccos(cos 7π ) = 6 5π 64. cos−1 (cos 7π ) = 416 of 26
  48. 48. The Inverse Cosine FunctionDetermine the following values:1. cos−1 1 2 = π 3 π2. Arccos 0 = 23. Arccos(cos 7π ) = 6 5π 64. cos−1 (cos 7π ) = 4 π 416 of 26
  49. 49. Pick-up quiz: Quiz # 1Evaluate the following values: √ 21. Arcsin(− 2 ) √ 32. Arccos(− 2 )3. sin−1 (sin 2π ) 34. cos−1 (cos 11π ) 65. Arcsin(cos 5π ) 3 17 of 26
  50. 50. Pick-up quiz: Quiz # 1Evaluate the following values: √ 21. Arcsin(− 2 ) = −π 4 √ 3 5π2. Arccos(− 2 ) = 63. sin−1 (sin 2π ) = 3 π 34. cos−1 (cos 11π ) = 6 π 65. Arcsin(cos 5π ) = 3 π 6 17 of 26
  51. 51. Inverse Trigonometric Functions of Non-Special AnglesExample 1: Evaluate cos(sin−1 4 ) 5 Let θ = sin−1 4 518 of 26
  52. 52. Inverse Trigonometric Functions of Non-Special AnglesExample 1: Evaluate cos(sin−1 4 ) 5 Let θ = sin−1 4 5 sin θ =18 of 26
  53. 53. Inverse Trigonometric Functions of Non-Special AnglesExample 1: Evaluate cos(sin−1 4 ) 5 Let θ = sin−1 4 5 4 sin θ = 518 of 26
  54. 54. Inverse Trigonometric Functions of Non-Special AnglesExample 1: Evaluate cos(sin−1 4 ) 5 Let θ = sin−1 4 5 4 sin θ = 5 cos θ =18 of 26
  55. 55. Inverse Trigonometric Functions of Non-Special AnglesExample 1: Evaluate cos(sin−1 4 ) 5 Let θ = sin−1 4 5 4 sin θ = 5 3 cos θ = 518 of 26
  56. 56. Inverse Trigonometric Functions of Non-Special AnglesExample 1: Evaluate cos(sin−1 4 ) 5 Let θ = sin−1 4 5 4 sin θ = 5 3 cos θ = 5 cos θ > 0 because θ = sin−1 4 5 can only be in Q1 or Q4.18 of 26
  57. 57. Inverse Trigonometric Functions of Non-Special AnglesExample 2: Evaluate cos(sin−1 − 3 ) 2 Let θ = sin−1 − 2 319 of 26
  58. 58. Inverse Trigonometric Functions of Non-Special AnglesExample 2: Evaluate cos(sin−1 − 3 ) 2 Let θ = sin−1 − 2 3 sin θ =19 of 26
  59. 59. Inverse Trigonometric Functions of Non-Special AnglesExample 2: Evaluate cos(sin−1 − 3 ) 2 Let θ = sin−1 − 2 3 sin θ = − 2 319 of 26
  60. 60. Inverse Trigonometric Functions of Non-Special AnglesExample 2: Evaluate cos(sin−1 − 3 ) 2 Let θ = sin−1 − 2 3 sin θ = − 2 3 cos θ =19 of 26
  61. 61. Inverse Trigonometric Functions of Non-Special AnglesExample 2: Evaluate cos(sin−1 − 3 ) 2 Let θ = sin−1 − 2 3 sin θ = − 2 3 2 cos θ = 1 − −2 319 of 26
  62. 62. Inverse Trigonometric Functions of Non-Special AnglesExample 2: Evaluate cos(sin−1 − 3 ) 2 Let θ = sin−1 − 2 3 sin θ = − 2 3 √ 2 5 cos θ = 1 − −2 3 = 319 of 26
  63. 63. Inverse Trigonometric Functions of Non-Special AnglesExample 2: Evaluate cos(sin−1 − 3 ) 2 Let θ = sin−1 − 2 3 sin θ = − 2 3 √ 2 5 cos θ = 1 − −2 3 = 3 cos θ > 0 because θ = sin−1 2 3 can only be in Q1 or Q4.19 of 26
  64. 64. Inverse Trigonometric Functions of Non-Special AnglesExample 3: Evaluate cos sin−1 (− 2 ) + cos−1 ( 1 ) 3 620 of 26
  65. 65. Inverse Trigonometric Functions of Non-Special AnglesExample 3: Evaluate cos sin−1 (− 2 ) + cos−1 ( 1 ) 3 6 α = sin−1 (− 2 ) 3 β = cos−1 ( 6 ) 120 of 26
  66. 66. Inverse Trigonometric Functions of Non-Special AnglesExample 3: Evaluate cos sin−1 (− 2 ) + cos−1 ( 1 ) 3 6 α = sin−1 (− 2 ) 3 β = cos−1 ( 6 ) 1 sin α = − 2 3 cos β = 1 620 of 26
  67. 67. Inverse Trigonometric Functions of Non-Special AnglesExample 3: Evaluate cos sin−1 (− 2 ) + cos−1 ( 1 ) 3 6 α = sin−1 (− 2 ) 3 β = cos−1 ( 6 ) 1 sin α = − 2 3 cos β = 1 6 √ √ 5 35 cos α = 3 sin β = 620 of 26
  68. 68. Inverse Trigonometric Functions of Non-Special AnglesExample 3: Evaluate cos sin−1 (− 2 ) + cos−1 ( 1 ) 3 6 α = sin−1 (− 2 ) 3 β = cos−1 ( 6 ) 1 sin α = − 2 3 cos β = 1 6 √ √ 5 35 cos α = 3 sin β = 6 cos(α + β) = cos α cos β − sin α sin β20 of 26
  69. 69. Inverse Trigonometric Functions of Non-Special AnglesExample 3: Evaluate cos sin−1 (− 2 ) + cos−1 ( 1 ) 3 6 α = sin−1 (− 2 ) 3 β = cos−1 ( 6 ) 1 sin α = − 2 3 cos β = 1 6 √ √ 5 35 cos α = 3 sin β = 6 cos(α + β) = cos α cos β − sin α sin β √ √ 5 1 35 = 3 6 − −2 3 620 of 26
  70. 70. Inverse Trigonometric Functions of Non-Special AnglesExample 3: Evaluate cos sin−1 (− 2 ) + cos−1 ( 1 ) 3 6 α = sin−1 (− 2 ) 3 β = cos−1 ( 6 ) 1 sin α = − 2 3 cos β = 1 6 √ √ 5 35 cos α = 3 sin β = 6 cos(α + β) = cos α cos β − sin α sin β √ √ 5 1 35 = 3 6 − −2 3 6 √ √ 5 + 2 35 = 1820 of 26
  71. 71. The Inverse Tangent FunctionFinding the graph of f (x) = tan−1 x f (x) = tan x21 of 26
  72. 72. The Inverse Tangent FunctionFinding the graph of f (x) = tan−1 x f (x) = Tan x, x ∈ (− π , π ) 2 221 of 26
  73. 73. The Inverse Tangent FunctionFinding the graph of f (x) = tan−1 x f (x) = Tan x, x ∈ (− π , π ) 2 2 Restrict the domain so that it becomes one-to-one.21 of 26
  74. 74. The Inverse Tangent FunctionThe inverse of f (x) = Tan x, x ∈ (− π , π ) 2 2 f (x) = Tan x, x ∈ (− π , π ) 2 222 of 26
  75. 75. The Inverse Tangent FunctionThe inverse of f (x) = Tan x, x ∈ (− π , π ) 2 2 Find the graph of the inverse by flipping along the diagonal22 of 26
  76. 76. The Inverse Tangent FunctionThe inverse of f (x) = Tan x, x ∈ (− π , π ) 2 2 f (x) = tan−1 x = Arctan x = inverse tangent of x22 of 26
  77. 77. The Inverse Tangent FunctionThe inverse of f (x) = Tan x, x ∈ (− π , π ) 2 2 f (x) = tan−1 x = Arctan x = inverse tangent of x Domain: Range:22 of 26
  78. 78. The Inverse Tangent FunctionThe inverse of f (x) = Tan x, x ∈ (− π , π ) 2 2 f (x) = tan−1 x = Arctan x = inverse tangent of x Domain: x ∈ R Range: y ∈ (− π , π ) 2 222 of 26
  79. 79. Other Inverse Trigonometric Functions f (x) = cot x f (x) = sec x f (x) = csc x23 of 26
  80. 80. Other Inverse Trigonometric Functions f (x) = Cot x, x ∈ (0, π) f (x) = Sec x, x ∈ [0, π] f (x) = Csc x, x ∈ (− π , π ) 2 223 of 26
  81. 81. Other Inverse Trigonometric Functions f (x) = Cot x, x ∈ (0, π) f (x) = Sec x, x ∈ [0, π] f (x) = Csc x, x ∈ (− π , π ) 2 223 of 26
  82. 82. Other Inverse Trigonometric Functions f (x) = Arccot x f (x) = Arcsec x f (x) = Arccsc x23 of 26
  83. 83. Other Inverse Trigonometric Functions f (x) = Arccot x Domain: x ∈ R Range: {0 < y < π} f (x) = Arcsec x Domain: {x ≤ −1} ∪ {x ≥ 1} Range: {0 ≤ y ≤ π, y = π } 2 f (x) = Arccsc x Domain: {x ≤ −1} ∪ {x ≥ 1} Range: {− π ≤ y ≤ π , y = 0} 2 224 of 26
  84. 84. Ranges of the Inverse Trigonometric Functions25 of 26
  85. 85. Ranges of the Inverse Trigonometric Functions f (x) = Arcsin(x) f (x) = Arccos(x) f (x) = Arctan(x) f (x) = Arccot(x) f (x) = Arccsc(x) f (x) = Arcsec(x)25 of 26
  86. 86. Any questions?26 of 26

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