Graphs of y = a sin x and y = a cos x

The Sine Curve y = a sin t

Amplitude

The a in the expression y = a sin x represents the amplitude of the graph. It is an indication of how much energy the wave contains.

The amplitude is the distance from the "resting" position (otherwise known as the mean value or average value) of the curve. In the interactive above, the amplitude can be varied from 10 to 100 units.

Amplitude is always a positive quantity. We could write this using absolute value signs. For the curves y = a sin x, amplitude = a

Graph of Sine x - with varying amplitudes

We start with y = sin x.

It has amplitude = 1 and period = 2π.

Now let's look at the graph of y = 5 sin x.

This time we have amplitude = 5 and period = 2π. (I have used a different scale on the y-axis.)

And now for y = 10 sin x.

Amplitude = 10 and period = 2π.

For comparison, and using the same y-axis scale, here are the graphs of p(x) = sin x, q(x) = 5 sin x and r(x) = 10 sin x on the one set of axes.

Note that the graphs have the same period (which is 2π) but different amplitude.

Graph of Cosine x - with varying amplitudes

Now let's have a look at the graph of y = cos x.

We note that the amplitude = 1 and period = 2π.

Similar to what we did with y = sin x above, we now see the graphs of
p(x) = cos x
q(x) = 5 cos x
r(x) = 10 cos x

on one set of axes, for comparison:



Note: For the cosine curve, just like the sine curve, the period of each graph is the same (2π), but the amplitude has changed.