In this section, we will introduce the inverse trigonometric functions and construct their...

In this section, we will introduce the inverse trigonometric functions and construct their derivative formulas.

Section 3.4 Inverse Trigonometric Functions

Recall

A function is called one-to-one if whenever

it must be true that .

That is, an output value cannot come from two different input values.

Recall

A function is called one-to-one if whenever

it must be true that .

That is, an output value cannot come from two different input values.

This function is not one-to-one.

Recall

A function is called one-to-one if whenever

it must be true that .

That is, an output value cannot come from two different input values.

Significance:

Only one-to-one functions have inverse functions.

Inverse Sine Function

Conceptual Idea

Below is shown the graph of

This function is not one-to-one and so has no inverse function.

Inverse Sine Function

Conceptual Idea

Below is shown the graph of

This function has an inverse.

Consider restricting the domain of the sine function to:

This is the function in blue shown to the left.

Inverse Sine Function

Definition

The function is the inverse of the sine function with restricted domain .

That is, the arcsin(x) is the angle θ in the interval

with .

Inverse Cosine Function

Conceptual Idea

Below is shown the graph of

This function is not one-to-one and so has no inverse function.

Inverse Cosine Function

Conceptual Idea

Below is shown the graph of

This function has an inverse.

Consider restricting the domain of the cosine function to:

This is the function in blue shown to the left.

Inverse Cosine Function

Definition

The function is the inverse of the cosine function with restricted domain .

That is, the arccos(x) is the angle θ in the interval

with .

Inverse Tangent Function

Conceptual Idea

Below is shown the graph of

This function is not one-to-one and so has no inverse function.

Inverse Tangent Function

Conceptual Idea

Below is shown the graph of

This function has an inverse function.

Consider restricting the domain of the tangent function to:

This is the function in blue shown to the left.

Inverse Tangent Function

Definition

The function is the inverse of the tangent function with restricted domain .

That is, the arctan(x) is the angle θ in the interval

with .

Example 1

Use the definitions of the section to find the exact

value of .

Example 2

Use the definitions of the section to find the exact

value of .

Example 3

Use the definitions of the section to find the exact

value of .

Example 4

Use the definitions of the section to find the exact

value of .

TheoremDerivatives of Inverse Trig.

Functions

The following are true:

TheoremDerivatives of Inverse Trig.

Functions

The following are true:

Example 5

Find the derivative of the function .

Example 6

Find the derivative of the function .

Example 7

Find the derivative of the function .