Sec. 6.6: Inverse Trigonometric Functions

In this section, we will

I revisit definitions of trig functions - both right triangle andunit circle definitions

I define inverse trig functions and their domains and ranges

I revisit the cancellation laws given a function and its inversefunction

I find the derivatives of the inverse trig functions

I look at integral formulas of certain functions whose integralsare inverse trig functions

Definition of trig functions and the ratios of the sides

Unit circle definition

x = cos θ, y = sin θ

Example 1) Identify the corresponding (x , y) on the U.C. if θ = π6

Example 2) Find (x , y) if θ = 11π6

Graph of the Sine function

Here’s the graph of the sine function.

Since the function is not 1-1, we need to restrict its domain todefine its inverse function. One way it can be done is by restringthe domain to [−π/2, π/2]. That is, let

D(sin x) = [−π/2, π/2]

The graph of the inverse sine function

Here’s the graph of the sine function whose domain restricted to[−π/2, π/2].

For the above function, state its domain and range.

Recall that for the inverse function, we flip everything about y = x .

The graph of the inverse sine function (Continued)

What are the domain and the range of the inverse sine function?

Graph of the Cosine function

Here’s the graph of the cosine function.

To define its inverse function, we will need to restrict its domain,so that the cosine function becomes 1-1. One way it can be doneis by restring the domain to [0, π]. So, let

D(cos x) = [0, π].

The graph of the inverse cosine functionHere’s the graph of the cosine function with its domain restrictedto [0, π].

For the above function, what is its domain and range?

Recall that for the inverse function, we flip everything about y = x .

The graph of the inverse cosine function (Continued)

What are the domain and the range of the inverse cosine function?

Cancellation equationsGiven f (x) with its domain D(f (x)), if g(x) = f −1(x), then

I f −1(f (x)) = x for x ∈ D(f )I f (f −1(x)) = x for x ∈ D(f −1)

Examples) Recall that

1. e(ln x) = x for x ∈ D(ln x)

2. ln(ex) = x for x ∈ D(ex)

3. a( ) = x for x ∈

4. loga( ) = x for x ∈

Cancellation equations for the inverse sine function

Theorem

1. sin−1(sin x) = x , −π2 ≤ x ≤ π

2

2. sin(sin−1 x) = x , −1 ≤ x ≤ 1

Example) Evaluate 1) sin−1(1/2), 2) tan(arcsin 13), 3) sin−1(2).

Cancellation equations for the inverse cosine function

Theorem

1. cos−1(cos x) = x , 0 ≤ x ≤ π2. cos(cos−1 x) = x , −1 ≤ x ≤ 1

Example) Evaluate 1) cos−1(1/2), 2) tan(arccos 14), 3)

cos−1(−2).

Graph of the tangent function

Graph of the inverse tangent function

1. What are the domain and the range of the inverse tangentfunction?

2. limx→∞ tan−1(x) =3. limx→−∞ tan−1(x) =

Derivatives of the inverse trig functions : arcsine

Theorem

d

dxsin−1(x) =

1√1− x2

, −1 < x < 1

Example) Given f (x) = sin−1(2x), find

1. Domain of f2. f

′(x)

3. Domain of f′

Derivatives of the inverse trig functions: arccosine

Theorem

d

dxcos−1(x) = − 1√

1− x2, −1 < x < 1

Example) Given f (x) = cos−1(x2 − 1), find

1. Domain of f2. f

′(x)

3. Domain of f′

Derivatives of the inverse trig functions: arctangent

Theorem

d

dxtan−1(x) =

1

1 + x2, −∞ < x <∞

Example) Given f (x) = tan−1(3x), find

1. Domain of f2. f

′(x)

3. Domain of f′

Integral formulas

Theorem

1.∫

1√1−x2 dx = sin−1 x + C

2.∫− 1√

1−x2 dx = cos−1 x + C

3.∫

11+x2

dx = tan−1 x + C

Examples) Evaluate

1.∫ 1/20

1√1−x2 dx

2.∫ 1/40

1√1−4x2 dx

3.∫ 10

41+x2

dx

Examples1. Verify that ∫

1

a2 + x2dx =

1

atan−1(

x

a) + C

2. Find∫

19+x2

dx

3. Find∫ 30

19+x2

dx

4. Find∫

x9+x4

dx