# 2. Graphs of *y *=* a* sin *bx* and *y *=* a*
cos *bx*

by M. Bourne

The variable *b* in both of the graph types

*y*=*a*sin*bx**y*=*a*cos*bx*

affects the **period** (or wavelength) of the graph. The **period** is the distance (or time) that it takes for the sine or cosine curve to begin repeating again.

## Interactive - Pistons and the Period of a Sine Curve

Here's an applet that you can use to explore the concept of **period** and **frequency** of a sine curve. The `"frequency" = 1/"period"`. We'll see more on this below.

The piston engine is the most commonly used engine in the world. Its motion can be described using a sine curve.

### Things to Do

- Click "Start" and you will see a sine curve traced out as the piston goes up and down.
- Now, change the value of
*b*using the slider. If you increase*b*, the period for each cycle will go down and the frequency will increase. - When `b` is not an integer, some of the curve at the beginning is not shown, so that the motion is smooth.
- When `b < 1`, some of the curve "disappears" off the end, since the period is greater than `2pi`.
- Observe the number of cycles that you see between
*t*= 0 and*t*= 2π. For*b*= 1 you see one cycle, for*b*= 2, you see 2 cycles, and so on. If*b*< 1, you do not see any complete cycles.

The units along the horizontal (time) axis are in radians. So *π* = 3.14 radians and 2*π* = 6.28 radians.

Frame rate = 0 fps

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## Did you notice?

- The variable `b` gives the number of cycles between `0` and `2pi`.
- Higher `b` gives higher frequency (and lower period).

### Formula for Period

The relationship between `b` and the **period** is given by:

`"Period"=(2pi)/b`

**Note:** As *b* gets larger, the period
decreases.

## Changing the Period

Now let's look at some still graphs to see what's going on.

The graph of *y *= 10 cos *x*, which we learned about in the last section, sine and cosine curves, is as follows.

As we learned, the **period** is `2pi`.

Next we see *y *= 10 cos 3*x*. Note the `3` inside the cosine term.

Notice that the period is different. (However, the amplitude is `10` in each example.)

This time the curve starts to repeat itself at `x=(2pi)/3`. This is consistent with the formula we met above:

`"Period"=(2pi)/b`

Now let's view the 2 curves on the same set of axes. Note that both graphs have an amplitude of `10` units, but their period is different.

### Good to Know...

**Tip 1:** The number *b* tells us the number of cycles in each
2*π*.

For *y *= 10 cos *x*, there is **one** cycle
between `0` and 2*π* (because *b* = 1).

For *y *= 10 cos 3*x*, there are **3** cycles
between `0` and 2*π* (because *b* = 3).

**Tip 2: **Remember, we are now operating using RADIANS. Recall that:

2

π= 6.283185...

and that

2

π= 360°

We only use **radians** in this chapter.

For a reminder, go to: Radians

### Exercises

### Need Graph Paper?

**1.** Sketch 2 cycles of *y* = 3 cos 8*x*.

[You can use the Java applet above to help you understand how the sketch works.)

**2.** Sketch 2 cycles of *y* = cos 10*x*.

**3. **Sketch 2 cycles of *y* = 5 sin 2*π**x*.

4. Sketch 2 cycles of `y = 4\ sin\ x/3`

## Defining Sine Curves using Frequency

It is common in electronics to express the sin graph in terms of the frequency *f* as follows:

y= sin 2πft

This is very convenient, since we don't have to do any calculation to find the frequency (like we were doing above). The frequency, *f*, is normally measured in **cycles/second**, which is the same as Hertz (**Hz**).

The **period** of the curve (the time it takes to go from one crest to the next crest) can be found easily once we know the frequency:

`T=1/f`

The units for period are normally **seconds**.

### Example

Household voltage in the UK is alternating current, `240\ "V"` with frequency `50\ "Hz"`. What is the equation describing this voltage?

## Coming Next

In the next section we learn about phase shift.

See Graphs of y = a sin (bx + c).

Later, we learn some Applications of Trigonometric Graphs.

But first, let's see another application of frequency.

## Music Example

The frequency of a note in music depends on the **period** of the wave. If the frequency is high, the period is short; if the frequency is low, the period is longer.

A student recently asked me an interesting question. She wanted to know the frequencies of all the notes on a piano.

A piano is tuned to A = 440 Hz (cycles/second) and the other notes are evenly spaced, 12 notes to each octave. A note an octave higher than A = 440 Hz has twice the frequency (880 Hz) and an octave lower than A = 440 Hz has half the frequency (`220\ "Hz"`).

Click here to find out the frequencies of notes on a piano.

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