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
x y 1
2
2
Definitions of Trigonometric Functions
Let 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
t 

3
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)
cos (-t) = cos (t)
tan (-t) = -tan (t)
sec (-t) = sec (t)
csc (-t) = -csc (t)
cot (-t) = -cot (t)
Evaluating Trigonometric Functions with a
Calculator (Mode -> Radians)