Section 4.2 Trigonometric Functions: The Unit Circle

What you should learn:

• Identify a unit circle and describe its relationship to real numbers.

• Evaluate trigonometric functions using the unit circle.

• Use the domain and period to evaluate sine and cosine functions.

• Use a calculator to evaluate trigonometric functions.

Unit Circle

2 2 1x y

Definitions of Trigonometric FunctionsLet t be a real number and let (x, y) be a point on the unit circle corresponding to t.

sin t = y

cos t = x

tan t = y/x, x ≠ o

cot t = x/y, y ≠ o

sec t = 1/x, x ≠ o

csc t = 1/y, y ≠ o

Example 1: Evaluating Trigonometric Functions.

Example 2: Evaluating Trigonometric Functions.Evaluate the six trigonometric functions for

3t

Domain and Period of Sine and Cosine

The domain of the sine and cosine functions is the set of real numbers. The range of the functions is from -1 to 1.

Definition of Periodic Function

A function f is periodic if there exists a positive real number c such that

f( t + c) = f(t)

for all t in the domain of f.

Odd Functions Even Functions

sin (-t) = -sin (t)

tan (-t) = -tan (t)

csc (-t) = -csc (t)

cot (-t) = -cot (t)

cos (-t) = cos (t)

sec (-t) = sec (t)

Evaluating Trigonometric Functions with a Calculator (Mode -> Radians)