Section 4.5

Graphs of Sine and Cosine

Overview

• In this section we first graph y = sin x andy = cos x.

• Then we graph transformations of sin x andcos x by applying changes in amplitude, period, and phase shift.

Some helpful information

• The period of both sine and cosine is 2π. This means that, after 2π the values of sine and cosine begin to repeat themselves.

• The domain of both sine and cosine is “all real numbers”.

• The range of both sine and cosine is [-1,1].

The graphs of sine and cosine

• We will use our completed table to get the necessary ordered pairs.

• The x-coordinates are the angles, measured in radians.

• The y-coordinates are the values of sine or cosine, depending on the function being graphed.

• Note particularly what the graphs do at 0, π/2, π, 3π/2, and 2π. These five points are key because they occur at the beginning, ¼ of the way, ½ of the way, ¾ of the way, and back at the beginning.

Sinusoidal Graphs

• Graphs of sine and cosine are generally referred to as sinusoidal graphs.

• The general form of the equations are

)cos(

)sin(

CBxAy

CBxAy

The Amplitude

• Amplitude refers to how high and how low the graphs goes.

• The amplitude of a sinusoidal graph is |A|.

Examples

• Find the amplitude of each of the following:

xy

xy

cos5

sin7

The Period

• Period refers to the length of the graph before the values start to repeat themselves.

• The period of a sinusoidal graph is given byB

2

Examples

• Find the period of each of the following:

xy

xy

4sin6

8sin

Phase Shift

• Phase shift refers to a horizontal translation of all the points on the graph.

• The value of the phase shift is x =• Look at the sign in your parentheses: if the

sign is (+), shift the graph to the left. If the sign is (-), shift the graph to the right.

B

C

Five Steps

1. Identify the amplitude, the period, and the phase shift.

2. Divide the period into four equal parts. Use ¼, ½, and ¾ to get the values of x.

3. If the phase shift is to the left, subtract from each of the x-values. If the phase shift is to the right, add to each of the x-values.

4. Note the behavior of the graph at each x-value (intercept, maximum, or minimum). Be sure to take amplitude into account. Plot the points.

B

C

B

C

Examples

• Find the phase shift of each of the following:

)23sin(2

1

)22cos(3

4cos3

1

)4sin(

xy

xy

xy

xy