## 2.9 Derivatives of the Trigonometric Functions

### Summary

The basic differentiation formulas for each of the trigonometric functions are introduced. Only the derivative of the sine function is computed directly from the limit definition. The derivatives of all the other trig functions are derived by using the general differentiation rules.

By the end of your studying, you should know:

• The limit of sin(x)/x as x goes to 0.
• How to use the limit above to compute the limit of related quotients.
• The derivatives of sine, cosine, tangent, secant, cosecant, and cotangent.
• How to find the derivative of any function involving a trig function and application of the general rules of differentiation.

On-screen applet instructions: Use the slider to let x → 0. What is the limit of sin(x)/x as x → 0?

### Examples

Consider the picture below:

θ is the central angle of the circle, s is the arc intercepted by θ, and d is the chord defined by θ. Find

A pilot flying at 3 miles above the ground at 600 miles per hour sights the airport with a spotting scope. How fast must she turn the scope when the angle between the path and plane is 40o to keep the scope pointed at the airport?

A block at the end of a spring is stretched past its rest position and released. Its position at time t is given by the formula d(t) = 4cos(t). Find the velocity of the spring at time t. When does the block move fastest?

### Applets

Limit of sin(x)/x as x approaches 0

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 2.9 Derivatives of the 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

Derivatives of Trig Functions
World Web Math
Visual Calculus

#### Interesting Application

Nothing yet has been found. Any ideas?

Software requirements: For best results viewing and interacting with this page, get the free software listed here.

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