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 2π = 6.28 radians.
b = 1
Graph: y = sin(θ)
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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:
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 3x. 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:
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 2π.
For y = 10 cos x, there is one cycle between `0` and 2π (because b = 1).
For y = 10 cos 3x, there are 3 cycles between `0` and 2π (because b = 3).
Tip 2: Remember, we are now operating using RADIANS. Recall that:
2π = 6.283185...
2π = 360°
We only use radians in this chapter.
For a reminder, go to: Radians
Need Graph Paper?
1. Sketch 2 cycles of y = 3 cos 8x.
[You can use the Java applet above to help you understand how the sketch works.)
2. Sketch 2 cycles of y = cos 10x.
3. Sketch 2 cycles of y = 5 sin 2πx.
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:
The units for period are normally seconds.
Household voltage in the UK is alternating current, `240\ "V"` with frequency `50\ "Hz"`. What is the equation describing this voltage?
In the next section we learn about phase shift.
Later, we learn some Applications of Trigonometric Graphs.
But first, let's see another application of frequency.
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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