We can see that the graph of y=\cos x can simply be obtained by y=\sin x shifted \frac{\pi}{2} to the left. This is a translation, and the identity is not unfamiliar to us: \sin x=\cos \left(x-\frac{\pi}{2}\right).

Now we shall look at graphs of simple transformations of these graphs. We shall start with y=a \sin (n t)and y=b \cos (n t). Naturally, n would be a dilation from the y-axis, anda (and b) is a dilation from the x-axis. The mechanism is similar, and is explained in A1 – Functions and Graphs/Relationship of Transformations . Here we will only provide a few simple examples.