Period and Frequency of Sine and Cosine

Introduction: In this lesson, the period and frequency of basic graphs of sine and cosine will be discussed and illustrated.

y = sin(x) and y = cos(x) are periodic functions because all possible y values repeat in the same sequence over a given set of x values. The “length” of this interval of x values is called the period.

Physics Connections

In physics texts, these periodic, sinusoidal graphs are generally divided into two distinct categories determined by the units used on the x-axis. Each category has a specific vocabulary. Graphs with equations of the form: y = sin(t) or y = cos(t) are generally called vibration graphs. On these graphs the time needed along the x-axis for one oscillation or vibration is called the period. Graphs with equations of the form: y = sin(x) or y = cos(x) are generally called waveform graphs. On these graphs the distance along the x-axis that is required for one oscillation or vibration is called a wavelength.

For example, if y = sin(x) the graph of this classic wave repeats over a length of

along the x-axis.

We see the same wave over and over for all real numbers x. In the graph above, you can see three complete waves.

In this graph the WINDOW is X:

and Y: (-2, 2, 1).

Because three complete waves are shown in a distance of , the length of one wave is

making

the period of y = sin(x). The frequency of this graph is f =

Stated another way,

is the distance required along the x-axis to graph one complete wave. This means that one (1) wave will be completed every

units along the x-axis.

The frequency is the reciprocal of the period.

Physics Connections

In physics texts, frequency is also the reciprocal of period. But it has the units of hertz, or oscillation/second not the dimensionless expression 1/radians. Frequency would only be applied to vibration graphs having equations of the form y = sin(t) and y = cos(t).

1. What are the period and frequency of y = sin(2x)?

The 2 has the effect of shortening the wave length or period. Waves appear on the graph twice as frequently as in y = sin(x). The graph shown below uses a WINDOW of X:

and Y: (-2, 2, 1).

There are 6 complete waves in a distance along the x-axis of . Therefore the period or length of one wave will be

while the frequency, or the reciprocal of the period, will be .

2. What are the period and frequency of y = cos(3x)?

The 3 has the effect of making waves appear on the graph three times as often as y = cos(x). The graph shown below uses a WINDOW of X:

and Y: (-2, 2, 1).

There are 9 complete waves in a distance along the x-axis of

making the period . Note that

as shown on the graph.

The frequency is the reciprocal of the period or .

Generalizing: For either y = sin(Bx) or y = cos(Bx) the period is . If we represent the period with the variable P, we can use the following two relationships .

Physics Connections

In physics texts: period is represented by the variable T frequency is represented by the variable f where f = 1/T angular frequency, the number of radians per second for a rotating system, is represented by the variable omega, w, where w = 2pf

3. What are the period and frequency of y = ?

Using the formula

we have period = . This graph shown below uses the WINDOW X: (-2, 4, 1) and Y: (-2, 2, 1).

Notice that there are three complete waves in a distance along the x-axis of [4 - (-2)] = 6. The period can be seen from the graph as

and the frequency equals .

4. If the period of a sine function is , what is its equation? Describe how its graph looks.

Using the formula

we have . Allowing us to determine that the equation is . The WINDOW for the graph shown below is X:

and Y: (-2, 2, 1).

Although it is somewhat difficult to count the number of waves on this graph, there are a total of

along the x-axis in a distance of . The function's period of

can be determined by dividing: .

4. If the period of a cosine function is 3, what is the equation? Describe how its graph looks.

Using the formula , we have . This allows us to write the requested function's equation as . The graph shown below uses a WINDOW of X: (-1.5, 4.5, 0.25) and Y: (-2, 2, 1).

Since 2 complete waves are shown in a distance of [4.5 - (-1.5)] = 6, we can see from the graph that the period will be .