1. Inverse Trigonometric Functions
Mathematics 4
October 24, 2011
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2. Inverse Trigonometric Functions
3
If sin x = 5 , what is x?
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3. Inverse Trigonometric Functions
3
If sin x = 5 , what is x?
How do we isolate x from the equation above?
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4. Inverse Trigonometric Functions
Let us recall inverses!
• f (x) = y = 2x − 1
•
•
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5. Inverse Trigonometric Functions
Let us recall inverses!
• f (x) = y = 2x − 1
• f −1 (x) →
•
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6. Inverse Trigonometric Functions
Let us recall inverses!
• f (x) = y = 2x − 1
• f −1 (x) → x = 2y − 1 The variables are interchanged.
•
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7. Inverse Trigonometric Functions
Let 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
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8. Inverse Trigonometric Functions
Let us recall inverses!
Given the graph of g(x):
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9. Inverse Trigonometric Functions
Let us recall inverses!
The inverse of g(x) can be flipping the graph along the diagonal:
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10. Inverse Trigonometric Functions
Let us recall inverses!
This is the graph of g −1 (x)
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11. Inverse Trigonometric Functions
Does the function f (x) = sin x have an inverse?
f (x) = sin x
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12. Inverse Trigonometric Functions
Does the function f (x) = sin x have an inverse?
f (x) = sin x
No!
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13. Inverse Trigonometric Functions
Does 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!
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14. Inverse Trigonometric Functions
Does 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!
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15. Inverse Trigonometric Functions
Do any of the six trigonometric functions have inverses?
f (x) = sin x
f (x) = cos x
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16. Inverse Trigonometric Functions
Do any of the six trigonometric functions have inverses?
f (x) = tan x
f (x) = cot x
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17. Inverse Trigonometric Functions
Do any of the six trigonometric functions have inverses?
f (x) = sec x
f (x) = csc x
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18. Inverse Trigonometric Functions
How can we isolate x in f (x) = sin x if f (x) is not one-to-one?
f (x) = sin x
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19. Inverse Trigonometric Functions
How 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
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20. Inverse Trigonometric Functions
How 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.
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21. Inverse Trigonometric Functions
The inverse of f (x) = Sin x, x ∈ [− π , π ]
2 2
f (x) = Sin x, x ∈ [− π , π ]
2 2
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22. Inverse Trigonometric Functions
The inverse of f (x) = Sin x, x ∈ [− π , π ]
2 2
Find the graph of the inverse by flipping along the diagonal
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23. Inverse Trigonometric Functions
The inverse of f (x) = Sin x, x ∈ [− π , π ]
2 2
f (x) = sin−1 x = Arcsin x = inverse sine of x
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24. The Inverse Sine Function
Properties of f (x) = sin−1 x:
Domain:
Range:
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25. The Inverse Sine Function
Properties of f (x) = sin−1 x:
Domain: x ∈ [−1, 1]
Range:
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26. The Inverse Sine Function
Properties of f (x) = sin−1 x:
Domain: x ∈ [−1, 1]
Range: y ∈ [− π , π ]
2 2
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27. The Inverse Sine Function
Properties of f (x) = sin−1 x:
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28. The Inverse Sine Function
Determine the following values:
1. sin−1 1
2 =
2. Arcsin 1 =
3. sin−1 (sin π ) =
4
4. Arcsin(sin 7π ) =
6
5. sin−1 (sin 4π ) =
3
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29. The Inverse Sine Function
Determine the following values:
1. sin−1 1
2 = π
6
2. Arcsin 1 =
3. sin−1 (sin π ) =
4
4. Arcsin(sin 7π ) =
6
5. sin−1 (sin 4π ) =
3
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30. The Inverse Sine Function
Determine the following values:
1. sin−1 1
2 = π
6
π
2. Arcsin 1 = 2
3. sin−1 (sin π ) =
4
4. Arcsin(sin 7π ) =
6
5. sin−1 (sin 4π ) =
3
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31. The Inverse Sine Function
Determine the following values:
1. sin−1 1
2 = π
6
π
2. Arcsin 1 = 2
3. sin−1 (sin π ) =
4
π
4
4. Arcsin(sin 7π ) =
6
5. sin−1 (sin 4π ) =
3
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32. The Inverse Sine Function
Determine the following values:
1. sin−1 1
2 = π
6
π
2. Arcsin 1 = 2
3. sin−1 (sin π ) =
4
π
4
4. Arcsin(sin 7π ) = − π
6 6
5. sin−1 (sin 4π ) =
3
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33. The Inverse Sine Function
Determine the following values:
1. sin−1 1
2 = π
6
π
2. Arcsin 1 = 2
3. sin−1 (sin π ) =
4
π
4
4. Arcsin(sin 7π ) = − π
6 6
5. sin−1 (sin 4π ) = − π
3 3
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34. The Inverse Cosine Function
Graphing the inverse cosine function
f (x) = cos x
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35. The Inverse Cosine Function
Graphing the inverse cosine function
f (x) = Cos x, x ∈ [0, π]
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36. The Inverse Cosine Function
Graphing the inverse cosine function
f (x) = Cos x, x ∈ [0, π]
Restrict the domain so that it becomes one-to-one.
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37. The Inverse Cosine Function
The inverse of f (x) = Cos x, x ∈ [0, π]
f (x) = Cos x, x ∈ [0, π]
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38. The Inverse Cosine Function
The inverse of f (x) = Cos x, x ∈ [0, π]
Find the graph of the inverse by flipping along the diagonal
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39. The Inverse Cosine Function
The inverse of f (x) = Cos x, x ∈ [0, π]
f (x) = cos−1 x = Arccos x = inverse cosine of x
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40. The Inverse Cosine Function
Properties of f (x) = cos−1 x:
Domain:
Range:
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41. The Inverse Cosine Function
Properties of f (x) = cos−1 x:
Domain: x ∈ [−1, 1]
Range:
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42. The Inverse Cosine Function
Properties of f (x) = cos−1 x:
Domain: x ∈ [−1, 1]
Range: y ∈ [0, π]
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43. The Inverse cosine Function
Properties of f (x) = cos−1 x:
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44. The Inverse Cosine Function
Determine the following values:
1. cos−1 1
2 =
2. Arccos 0 =
3. Arccos(cos 7π ) =
6
4. cos−1 (cos 7π ) =
4
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45. The Inverse Cosine Function
Determine the following values:
1. cos−1 1
2 = π
3
2. Arccos 0 =
3. Arccos(cos 7π ) =
6
4. cos−1 (cos 7π ) =
4
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46. The Inverse Cosine Function
Determine the following values:
1. cos−1 1
2 = π
3
π
2. Arccos 0 = 2
3. Arccos(cos 7π ) =
6
4. cos−1 (cos 7π ) =
4
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47. The Inverse Cosine Function
Determine the following values:
1. cos−1 1
2 = π
3
π
2. Arccos 0 = 2
3. Arccos(cos 7π ) =
6
5π
6
4. cos−1 (cos 7π ) =
4
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48. The Inverse Cosine Function
Determine the following values:
1. cos−1 1
2 = π
3
π
2. Arccos 0 = 2
3. Arccos(cos 7π ) =
6
5π
6
4. cos−1 (cos 7π ) =
4
π
4
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49. Pick-up quiz: Quiz # 1
Evaluate the following values:
√
2
1. Arcsin(− 2 )
√
3
2. Arccos(− 2 )
3. sin−1 (sin 2π )
3
4. cos−1 (cos 11π )
6
5. Arcsin(cos 5π )
3
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50. Pick-up quiz: Quiz # 1
Evaluate the following values:
√
2
1. Arcsin(− 2 ) = −π
4
√
3 5π
2. Arccos(− 2 ) = 6
3. sin−1 (sin 2π ) =
3
π
3
4. cos−1 (cos 11π ) =
6
π
6
5. Arcsin(cos 5π ) =
3
π
6
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51. Inverse Trigonometric Functions of Non-Special Angles
Example 1: Evaluate cos(sin−1 4 )
5
Let θ = sin−1 4
5
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52. Inverse Trigonometric Functions of Non-Special Angles
Example 1: Evaluate cos(sin−1 4 )
5
Let θ = sin−1 4
5
sin θ =
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53. Inverse Trigonometric Functions of Non-Special Angles
Example 1: Evaluate cos(sin−1 4 )
5
Let θ = sin−1 4
5
4
sin θ = 5
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54. Inverse Trigonometric Functions of Non-Special Angles
Example 1: Evaluate cos(sin−1 4 )
5
Let θ = sin−1 4
5
4
sin θ = 5
cos θ =
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55. Inverse Trigonometric Functions of Non-Special Angles
Example 1: Evaluate cos(sin−1 4 )
5
Let θ = sin−1 4
5
4
sin θ = 5
3
cos θ = 5
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56. Inverse Trigonometric Functions of Non-Special Angles
Example 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.
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57. Inverse Trigonometric Functions of Non-Special Angles
Example 2: Evaluate cos(sin−1 − 3 )
2
Let θ = sin−1 − 2
3
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58. Inverse Trigonometric Functions of Non-Special Angles
Example 2: Evaluate cos(sin−1 − 3 )
2
Let θ = sin−1 − 2
3
sin θ =
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59. Inverse Trigonometric Functions of Non-Special Angles
Example 2: Evaluate cos(sin−1 − 3 )
2
Let θ = sin−1 − 2
3
sin θ = − 2
3
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60. Inverse Trigonometric Functions of Non-Special Angles
Example 2: Evaluate cos(sin−1 − 3 )
2
Let θ = sin−1 − 2
3
sin θ = − 2
3
cos θ =
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61. Inverse Trigonometric Functions of Non-Special Angles
Example 2: Evaluate cos(sin−1 − 3 )
2
Let θ = sin−1 − 2
3
sin θ = − 2
3
2
cos θ = 1 − −2
3
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62. Inverse Trigonometric Functions of Non-Special Angles
Example 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
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63. Inverse Trigonometric Functions of Non-Special Angles
Example 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.
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64. Inverse Trigonometric Functions of Non-Special Angles
Example 3: Evaluate cos sin−1 (− 2 ) + cos−1 ( 1 )
3 6
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65. Inverse Trigonometric Functions of Non-Special Angles
Example 3: Evaluate cos sin−1 (− 2 ) + cos−1 ( 1 )
3 6
α = sin−1 (− 2 )
3 β = cos−1 ( 6 )
1
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66. Inverse Trigonometric Functions of Non-Special Angles
Example 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
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67. Inverse Trigonometric Functions of Non-Special Angles
Example 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
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68. Inverse Trigonometric Functions of Non-Special Angles
Example 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
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69. Inverse Trigonometric Functions of Non-Special Angles
Example 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
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70. Inverse Trigonometric Functions of Non-Special Angles
Example 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
=
18
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71. The Inverse Tangent Function
Finding the graph of f (x) = tan−1 x
f (x) = tan x
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72. The Inverse Tangent Function
Finding the graph of f (x) = tan−1 x
f (x) = Tan x, x ∈ (− π , π )
2 2
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73. The Inverse Tangent Function
Finding 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.
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74. The Inverse Tangent Function
The inverse of f (x) = Tan x, x ∈ (− π , π )
2 2
f (x) = Tan x, x ∈ (− π , π )
2 2
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75. The Inverse Tangent Function
The inverse of f (x) = Tan x, x ∈ (− π , π )
2 2
Find the graph of the inverse by flipping along the diagonal
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76. The Inverse Tangent Function
The inverse of f (x) = Tan x, x ∈ (− π , π )
2 2
f (x) = tan−1 x = Arctan x = inverse tangent of x
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77. The Inverse Tangent Function
The inverse of f (x) = Tan x, x ∈ (− π , π )
2 2
f (x) = tan−1 x = Arctan x = inverse tangent of x
Domain: Range:
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78. The Inverse Tangent Function
The 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 2
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79. Other Inverse Trigonometric Functions
f (x) = cot x
f (x) = sec x
f (x) = csc x
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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 2
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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 2
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82. Other Inverse Trigonometric Functions
f (x) = Arccot x
f (x) = Arcsec x
f (x) = Arccsc x
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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 2
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84. Ranges of the Inverse Trigonometric Functions
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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)
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86. Any questions?
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