Section 4.6: Inverse Trigonometric Functions
Since the trigonometric functions are not one-to-one, they do not have inverse functions.
However, each of the trigonometric functions is one-to-one on a restricted domain, and inverse
functions can be defined for these restricted trigonometric functions.
The sine function, f (x) = sin x, is one-to-one on the restricted domain, −π/2 ≤ x ≤ π/2.
Therefore, the inverse sine function or arcsine function exists and is denoted by
y = sin−1 x
or
y = arcsin x.
h π πi
The domain of sin−1 x is [−1, 1] and the range is − , .
2 2
Example: Find the exact value of each expression.
1
−1
(a) sin
2
(b) sin−1
√ !
3
−
2
1
−1 1
Example: Find the exact value of tan sin
.
3
Note: Since y = sin x and y = sin−1 x are inverse functions, we have the following cancellation
equations
• sin−1 (sin x) = x for −
π
π
≤x≤ ,
2
2
• sin(sin−1 x) = x for −1 ≤ x ≤ 1.
These cancellation equations are valid only when x lies in the indicated intervals.
Example: Find the exact value of each expression.
7
(a) sin arcsin
10
π
(b) sin−1 sin
5
2
5π
(c) arcsin sin
4
Example: Use implicit differentiation to find the derivative of y = sin−1 x.
3
The cosine function, f (x) = cos x, is one-to-one on the restricted domain 0 ≤ x ≤ π.
Therefore, the inverse cosine function or arccossine function exists and is denoted by
y = cos−1 x
or
y = arccos x.
The domain of cos−1 x is [−1, 1] and the range is [0, π].
Example: Find the exact value of each expression.
1
−1
(a) cos
2
(b) arccos(1)
(c) cos−1
√ !
3
−
2
4
−1 4
Example: Find the exact value of tan cos
.
5
Note: Since y = cos x and y = cos−1 x are inverse functions, we have the following cancellation
equations
• cos−1 (cos x) = x for 0 ≤ x ≤ π
• cos(cos−1 x) = x for −1 ≤ x ≤ 1
These cancellation equations are valid only when x lies in the indicated intervals.
Example: Find the exact value of each expression.
(a) cos (arccos 0.8)
(b) cos
−1
3π
cos
4
5
4π
(c) arccos cos
3
Example: Use implicit differentiation to find the derivative of y = cos−1 x.
6
The tangent function, f (x) = tan x, is one-to-one on the restricted domain, −π/2 < x < π/2.
Therefore, the inverse tangent function or arctangent function exists and is denoted
by
y = tan−1 x
or
y = tan x.
π π
.
The domain of tan−1 x is (−∞, ∞) and the range is − ,
2 2
Example: Find the exact value of each expression.
(a) tan−1 (0)
(b) arctan(1)
√
(c) tan−1 ( 3)
7
Example: Find the exact value of sec (arctan 2).
Example: Evaluate each limit.
(a) lim arctan x
x→∞
(b) lim tan−1 x
x→−∞
−1
(c) lim sin
x→∞
x+1
2x + 1
8
Example: Use implicit differentiation to find the derivative of y = tan−1 x.
Theorem: (Derivatives of Inverse Trigonometric Functions)
d
1
(sin−1 x) = √
dx
1 − x2
1
d
(csc−1 x) = − √
dx
x x2 − 1
1
d
(cos−1 x) = − √
dx
1 − x2
1
d
(sec−1 x) = √
dx
x x2 − 1
d
1
(tan−1 x) = 2
dx
x +1
d
1
(cot−1 x) = − 2
.
dx
x +1
Example: Differentiate each function.
(a) f (x) = sin−1 (2x − 1)
9
(b) g(x) = x cos−1 x
(c) h(x) = tan−1 (sin x)
10