Chapter 4 Trigonometric Functions 1

4.2 The Unit Circle Objectives:  Evaluate trigonometric functions using the unit circle.  Use domain and period to evaluate sine and cosine functions.  Use a calculator to evaluate trigonometric functions. 2

What is the Unit Circle?  Equation of the unit circle: x 2 + y 2 = 1  Center: (0, 0)  Radius = 1 3

Unit Circle with Number Line  Imagine that the real number line is wrapped around the unit circle, as shown.  Note: the positive numbers wrap towards the positive y -axis and the negative numbers wrap towards the negative y -axis. 4

More Unit Circle  Each real number t corresponds to a point (x, y) on the circle.  Each real number t also corresponds to a central angle θ whose radian measure is t. 5

Compare Values (8 Segments) 6

Compare Values (12 Segments) 7

Definition of Trig Functions  Let t be a real number and let (x, y) be the point on the unit circle corresponding to t. Then the six trig functions are defined: 8

Example 1  Evaluate the six trig functions at each real number. 9

Exploration  Complete the activity (handout) in which you will investigate the periodic nature of the sine function as it relates to the unit circle. You will need a graphing calculator. 10

Sine and Cosine  Domain:  Range:  What happens when we add 2π to t ?  So, 11

In General  For n revolutions around the unit circle,  What is the period for sine and cosine? 12

Example 2  Evaluate using its period as an aid. 13

Even and Odd Functions  Even Function if f (–t) = f (t).  Odd Function if f (–t) = – f (t). 14

Our Friend, the Calculator  What do we need to always check before solving a trig problem with a calculator?  We can easily solve for sine, cosine, or tangent. How do we solve for cosecant, secant, and cotangent? 15

Homework 4.2  Worksheet