4.5 Graphs of Sine and Cosine Functions

The graph of a sine function is called a sine curve. Likewise the graph of a cosine function is called a cosine curve.

Lets take a look at the graph of 

The period (or wavelength) of a a sine curve is the how long it takes to complete one cycle of the graph. In the case of a sine curve the period is . The amplitude of a sine graph is the distance from the crest (or the maximum value) or the trough (the minimum value) to the mid line. The amplitude of a normal sine curve is 1. The range of this sine curve is [-1, 1].

Now lets take a look at the graph of 

Just like in the sine curve the period of this graph is . The amplitude, also like the sine graph, is 1. The range of this graph is also [-1,1]. As you can see there are many similarities between the sine and cosine curves. In fact, the cosine curve can be seen as the sine graph shifted to the left . Speaking of graphs getting shifted, lets take a look at transformations of sine and cosine graphs.

Transformations of a sine (or cosine graph)

The basic equation looks like this:

a= vertical stretch or a change in amplitude

      The amplitude of a function can be calculated by taking the absolute value of a -> 

      A change in amplitude has no affect of the period

      If a is negative then the graph is reflected in the x-axis

b= horizontal compression or stretch 

      In other words b changes the period

      A change in b has no affect on the amplitude

      The new period can be calculated then by plugging b into 

      If b<0 then the graph is reflected in the y-axis (which looks like a reflection in the x-axis)

c= a phase shift or a shift left or right

      As stated before the cosine graph can be seen as the sine graph shifted 

d= a vertical shift

     This is a shift of the mid line

Here is a link to a quick animation and interactive sine/cosine graph: Sine and Cosine Transformation Animation 

Sine curves and Cosine curves are related to the unit circle. One period is also one rotation around the unit circle, which is also . The x-values are the angles of the unit circle and the y-values are the corresponding value (whether it is cosine or sine) to that angle. The range of a normal sine graph and cosine graph is [-1,1] because on the unit circle, sine and cosine  values never exceed 1 or drop below -1.