## 4.10 Inverse Trigonometric Functions

### Summary

The inverse functions for the sine, cosine, and tangent are introduced. In defining them, issues are pointed out that must be considered in defining the inverse of any periodic function. Then the derivative of the inverse sine and the inverse tangent functions are derived. Their companion integration formulas give two new integrals that can be calculated from the Fundamental Theorem.

By the end of your studying, you should know:

• How to define the inverse sine (or arcsine) function.
• How to define the inverse cosine (or arccosine) function.
• How to define the inverse tangent (or arctangent) function.
• How to find values of the inverse functions.
• The derivative of the arcsine and arctangent functions, and the companion integration formulas.
• How to use these new derivatives and integrals in computing derivatives and integrals of related functions.

On-screen applet instructions: The applet allows you to experiment with the sine, cosine, and tangent functions to limit their domains for the purpose of defining their inverse functions. Use the pull-down menu to select the sine, cosine, or tangent. Then drag the mouse along the curve to identify a domain over which an inverse function can be defined.

### Examples

What is

What is

What is the derivative of

### Applets

Calculator: Values of Elementary Functions

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 4.10 Inverse Trigonometric Functions (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

### Resources on the Web

Information on Newton
Biographical data from St. Andrew's University's Web site
Excerpt from W.W. Rouse Ball's "A Short Account of the History of Mathematics"

Calculus Applications
Project Intermath

Inverse Trig Functions
Visual Calculus

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel