Warm-Up: 9/14/12 Find the amplitude, period, vertical asymptotes, domain, and range. Sketch the graph.

4.7 – Inverse Trigonometric Functions

In this section, you will learn to: Evaluate the inverse trigonometric functions Evaluate the composition of trigonometric functions

Functions:  In order for a relation to be a function, it must pass the vertical line test.  For a function to have an inverse, it must pass the horizontal line test.  Different values of x cannot yield the same values of y.

Function or not a function? 1) 2) 3)

Which function has an inverse? 1) 2) 3)

Inverse Sine Function: The sine function does not pass the horizontal test, therefore it does not have an inverse. However, if we restrict the domain, then it will pass the horizontal line test.

Inverse Sine Function:

Definition of an Inverse Sine Function:    

Graphing an Inverse Sine Function:  To sketch the graph of an inverse sine function, interchange the domain and the range of the original sine function.

Graphing an Inverse Sine Function:  Inverse functions are reflected about the line y = x.

Inverse Cosine Function: The cosine function does not pass the horizontal test, therefore it does not have an inverse. However, if we restrict the domain, then it will pass the horizontal line test.

Inverse Cosine Function:

Definition of an Inverse Cosine Function:    

Graphing an Inverse Cosine Function:  To sketch the graph of an inverse cosine function, interchange the domain and the range of the original cosine function.

Graphing an Inverse Cosine Function:  Inverse functions are reflected about the line

Inverse Tangent Function: The tangent function does not pass the horizontal test, therefore it does not have an inverse. However, if we restrict the domain, then it will pass the horizontal line test.

Inverse Tangent Function:

Inverse Tangent Function:

Inverse Tangent Function:

Definition of an Inverse Tangent Function:    

Graphing an Inverse Tangent Function:  To sketch the graph of an inverse tangent function, interchange the domain and the range of the original sine function.

Graphing an Inverse Tangent Function:  Inverse functions are reflected about the line

Find the exact value of the inverse functions:

Solutions:

Composition of Functions:

Composition of Function Examples: