Period and Frequency of Sine and CosineAlgebraLAB: Lessons

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.

The Lesson:

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 numbersx. In the graph above, you can see three complete waves.

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.

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).

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 .

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.

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 .

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 .

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: .

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 .