## 1.5 Trigonometric Functions

### Summary

The trigonometric functions are important for modeling periodic behavior. The circular function definitions of sine and cosine are introduced, as are the definitions of the tangent, the secant, the cosecant, and the cotangent functions. The trigonometric identities that follow readily from symmetry considerations are also developed.

By the end of your studying, you should know:

• How to define the radian measure of an angle.
• The definitions of the sine and cosine functions.
• The definitions of the other four basic trig functions.
• The graphs of the six basic trig functions.
• How to compute values of the trig functions for angles related to the standard reference triangles.
• How to graph or recognize stretches and/or shifts of the basic trig functions.
• Trigonometric identities that follow readily from the definitions of the trig functions, or from symmetry considerations.

On-screen applet instructions: The initial value of the angle is set to 1 (radian) in order to give a picture that is meaningful before there is any movement of the slider. The initial position of the slider is at angle 0, however, so that the graphs start at zero.

### Examples

Evaluate the following exactly:

A particle moving back and forth on the x axis has its position given as a function of time t as follows:

Where is the particle when t = 0 and how long does it take to return to this position?

Find all solutions to the following equation:

### Applets

Definitions of sin(x) and cos(x)
Trigonometric Identities

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 1.5 Trigonometric Functions (PDF).

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

#### Interesting Application

Do you see trig at work?

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