GRAPHS OF THE TRIGONOMETRIC FUNCTIONS

Topics in this lesson:
1. SINE GRAPHS
2. COSINE GRAPHS
3. SINE AND COSINE GRAPHS WITH PHASE SHIFTS
4. SECANT AND COSECANT GRAPHS
5. TANGENT GRAPHS
6. COTANGENT GRAPHS
1. SINE GRAPHS
Example Use the Unit Circle to graph two cycles of the function y sin x on the
interval [0, 4 ] .
Example Use the Unit Circle to graph two cycles of the function y sin x on the
interval [ 4 , 0] .
Definition The amplitude of a trigonometric function is one-half of the difference
between the maximum value of the function and the minimum value of the function
if the function has both of these values.
NOTE: The maximum value of a function is the largest y-coordinate on the graph
of the function and the minimum value of a function is the smallest y-coordinate
on the graph of the function if the graph has both of these values.
The sine and cosine functions will have an amplitude. However, the tangent,
cotangent, secant, and cosecant functions do not have an amplitude because these
functions do not have a maximum value nor a minimum value.
Definition The period of a trigonometric function is the distance needed to
complete one cycle of the graph of the function.
All the trigonometric functions have a period.
For the function y sin x , the amplitude of the function is 1 and the period is 2 .
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
Given the function y a sin b x , the amplitude of this function is a and the
period is
b
2
.
Theorem The sine function is an odd function. That is, sin ( ) sin for
all in the domain of the function.
NOTE: The domain of the sine function is all real numbers.
Examples Sketch two cycles of the graph of the following functions. Label the
numbers on the x- and y-axes.
1. y 5 sin 3x
Amplitude = 5 = 5 Period =
3
2
=
3
2
4
1
period =
3
2
4
1
=
2 3
1
=
6
y
5
x
6
3
2
3
2
6
5
6
7
3
4
x 5
NOTE: The first cycle begins