While working on an assignment about sound in your science class, your Instructor informs you that what you know as the "pitch" of a sound is, in fact, the frequency of the sound waves. He then plays a note on a musical instrument, and the pattern of the sound wave on a graph looks like this:

He then tells you to find the frequency of the sound wave from the graph? Can you do it?

### Period and Frequency

The **period** of a trigonometric function is the horizontal distance traversed before the

**Frequency** is a measurement that is closely related to period. In science, the frequency of a sound or light wave is the number of complete waves for a given time period (like seconds). In trigonometry, because all of these periodic functions are based on the unit circle, we usually measure frequency as the number of complete waves every

Period and frequency are inversely related. That is, the higher the frequency (more waves over

After observing the transformations that result from multiplying a number *in front of* the sinusoid, it seems natural to look at what happens if we multiply a constant *inside* the argument of the function, or in other words, by the

Notice that the number of waves for **2 waves** over the interval from 0 to *half* that distance—so the graph has been “scrunched” horizontally. The frequency of this graph is therefore 2, or the same as the constant we multiplied by in the argument. The period (the length for each complete wave) is

#### Finding the Period and Frequency

1. What is the frequency and period of

If we follow the pattern from the previous example, multiplying the angle by 3 should result in the sine wave completing a cycle **three times** as often as

This number that is multiplied by *decrease* the frequency, or multiply by a number that is less than 1. Remember that this dilation factor is *inversely* related to the period of the graph.

Adding, one last time to our equations from before, we now have:

2. What is the frequency and period of

Using the generalization above, the frequency must be

Thinking of it as a transformation, the graph is stretched horizontally. We would only see

3. What is the frequency and period of

Like the previous two problems, we can see that the frequency is

### Examples

#### Example 1

Earlier, you were asked to find the frequency of the sound wave from the graph.

By inspecting the graph

You can see that the wave takes about 6.2 seconds to make one complete cycle. This means that the frequency of the wave is approximately 1 cycle per second (since

#### Example 2

Draw a sketch of

The "2" inside the sine function makes the function "squashed" by a factor of 2 in the horizontal direction.

#### Example 3

Draw a sketch of \begin{align*}y = 2.5 \cos \pi x\end{align*} from 0 to \begin{align*}2\pi\end{align*}.

The \begin{align*}\pi\end{align*} inside the sine function makes the function "squashed" by a factor of \begin{align*}\pi\end{align*} in the horizontal direction.

#### Example 4

Draw a sketch of \begin{align*}y=4 \sin \frac{1}{2} x\end{align*} from 0 to \begin{align*}2\pi\end{align*}.

The \begin{align*}\frac{1}{2}\end{align*} inside the sine function makes the function "stretched" by a factor of \begin{align*}\frac{1}{2}\end{align*} in the horizontal direction.

### Review

Find the period and frequency of each function below.

- \begin{align*}y=\sin(4x)\end{align*}
- \begin{align*}y=\cos(2x)\end{align*}
- \begin{align*}y=\cos(\frac{1}{2}x)\end{align*}
- \begin{align*}y=\sin(\frac{3}{4}x)\end{align*}
- \begin{align*}y=\sin(3x)\end{align*}

Draw a sketch of each function from 0 to \begin{align*}2\pi\end{align*}.

- \begin{align*}y=\sin(3x)\end{align*}
- \begin{align*}y=\cos(5x)\end{align*}
- \begin{align*}y=3\cos(\frac{2}{5}x)\end{align*}
- \begin{align*}y=\frac{1}{2}\sin(\frac{3}{4}x)\end{align*}
- \begin{align*}y=-\sin(2x)\end{align*}
- \begin{align*}y=\tan(3x)\end{align*}
- \begin{align*}y=\sec(2x)\end{align*}
- \begin{align*}y=\csc(4x)\end{align*}

Find the equation of each function.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.15.