6.1 Inverse Circular Functions

One-to-One If each element of f maps to a different element in the range.

Horizontal Line Test

Any horizontal line will intersect the graph of a one-to-one function in at most one point.

If a function is one-to-one then it has an inverse function

Inverse Function

The inverse function of the one-to-one function f is defined as

Important Note!!

Summary of Inverse Functions

1. In a one-to-one function, each x-value corresponds to only one y-value and

each y-value corresponds to only one x-value.

2. If a function f is one-to-one, then f has an inverse function .

3. The domain of f is the range of , and the range of f is the domain of .

4. The graphs of f and are reflections of each other across the line .

5. To find from , follow these steps:

A. Replace with y and interchange x and y

B. Solve for y.

C. Replace y with

Inverse Sine Function

In order for sine x to be one-to-one, the domain must be restricted to and the range is [-1,1]. For the inverse function, the domain is [-1,1] and the range is .

Inverse Sine Function

Mental Thought

We can think of as

“y is the number (in radians) in the interval whose sine is x.

INVERSE SINE FUNCTION

Domain: [-1,1] Range:

● The inverse sine function is increasing and continuous on its domain [-1,1]

● Its x-intercept is 0, and its y-intercept is 0.

● Its graph is symmetric with respect to the origin; it is an odd function

Example

Find y in each equation.

a) b) c)

**Make sure the answer to an inverse function value is in the appropriate

range of the particular inverse function being considered!!

Inverse Cosine Function

Mental Thought

We can think of as

“y is the number (in radians) in the interval whose cosine is x.

INVERSE COSINE FUNCTION

Domain: [-1,1] Range:

● The inverse cosine function is decreasing and continuous on its domain [-1,1]

● Its x-intercept is 1, and its y-intercept is .

● Its graph is neither symmetric with respect to the y-axis nor the origin.

Examples

Find y in each equation.

a) b)

Inverse Tangent Function

Inverse Tangent Function

Domain: Range:

● The inverse tangent function is increasing and continuous on its domain

● Its x-intercept is 0, and its y-intercept is 0.

● Its graph is symmetric with respect to the origin; it is an odd function.

● The lines and are horizontal asymptotes.

Example

Find the degree measure of θ in the following:

The Remaining Inverse Circular Functions

Example

Find the degree measure of θ in the following:

Using a calculator

**Make sure calculator is in degree or radian mode as appropriate.

Find y in radians if

Do analytically and with Calculator

Be careful when finding inverse cotangent of a negative quantity. Because the calculator actually does inverse of tangent, the range must be adjusted

by adding or

Find θ in degrees if ans: 104.19^o

Examples

Evaluate each expression without a calculator.

a) b)

c) d)

e)