6. Inverse Trigonometric Functions
Let us recall inverses!
• f (x) = y = 2x − 1
• f −1 (x) → x = 2y − 1 The variables are interchanged.
•
3 of 26
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
3 of 26
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!
5 of 26
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!
5 of 26
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.
7 of 26
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
8 of 26
24. The Inverse Sine Function
Properties of f (x) = sin−1 x:
Domain:
Range:
9 of 26
25. The Inverse Sine Function
Properties of f (x) = sin−1 x:
Domain: x ∈ [−1, 1]
Range:
9 of 26
26. The Inverse Sine Function
Properties of f (x) = sin−1 x:
Domain: x ∈ [−1, 1]
Range: y ∈ [− π , π ]
2 2
9 of 26
27. The Inverse Sine Function
Properties of f (x) = sin−1 x:
10 of 26
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
11 of 26
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
11 of 26
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
11 of 26
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
11 of 26
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
11 of 26
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
11 of 26
34. The Inverse Cosine Function
Graphing the inverse cosine function
f (x) = cos x
12 of 26
35. The Inverse Cosine Function
Graphing the inverse cosine function
f (x) = Cos x, x ∈ [0, π]
12 of 26
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.
12 of 26
37. The Inverse Cosine Function
The inverse of f (x) = Cos x, x ∈ [0, π]
f (x) = Cos x, x ∈ [0, π]
13 of 26
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
13 of 26
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
13 of 26
40. The Inverse Cosine Function
Properties of f (x) = cos−1 x:
Domain:
Range:
14 of 26
41. The Inverse Cosine Function
Properties of f (x) = cos−1 x:
Domain: x ∈ [−1, 1]
Range:
14 of 26
42. The Inverse Cosine Function
Properties of f (x) = cos−1 x:
Domain: x ∈ [−1, 1]
Range: y ∈ [0, π]
14 of 26
52. Inverse Trigonometric Functions of Non-Special Angles
Example 1: Evaluate cos(sin−1 4 )
5
Let θ = sin−1 4
5
sin θ =
18 of 26
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
18 of 26
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 =
18 of 26
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
18 of 26
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.
18 of 26
58. Inverse Trigonometric Functions of Non-Special Angles
Example 2: Evaluate cos(sin−1 − 3 )
2
Let θ = sin−1 − 2
3
sin θ =
19 of 26
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
19 of 26
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 =
19 of 26
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
19 of 26
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
19 of 26
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.
19 of 26
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
20 of 26
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
20 of 26
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
20 of 26
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
20 of 26
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
20 of 26
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
20 of 26
71. The Inverse Tangent Function
Finding the graph of f (x) = tan−1 x
f (x) = tan x
21 of 26
72. The Inverse Tangent Function
Finding the graph of f (x) = tan−1 x
f (x) = Tan x, x ∈ (− π , π )
2 2
21 of 26
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.
21 of 26
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
22 of 26
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
22 of 26
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
22 of 26
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:
22 of 26
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
22 of 26
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
24 of 26
84. Ranges of the Inverse Trigonometric Functions
25 of 26
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