In an earlier section we already saw one relationship between sine and cosine, which came from using Pythagoras' theorem.
That gave us the result that
sin2(q)+cos2(q)=1.
This means that if we know the value of sine for some q, then we can work out the cosine using this formula. For example, remember one of the special angles was 450, or p/4 and that this was special because the opposite side and adjacent side were the same length. Well, that means that cos(450) and sin(450) must be the same. But from the formula above, that means that sin2(450) and cos2(450) must both be a half. So this gives us the result that we had before:
sin(450)=cos(450)=
There's another simple relationship between the sine and cosine functions. Did you notice that the curves of sine and cosine were very similar, both just a wave going up and down between 1 and -1, over and over again?
Well the only difference between the two curves is that one is shifted along a bit. Here they are together so you can see what I mean.
The curve that goes through 1 when q is zero is the cosine curve, since cos(0)=1. The other curve is sin(q), notice that sin(0)=0.
So, you can see that we could just shift one curve along a bit, well actually it's a distance of p/2 or 900, to get the other curve. This shows that the sine and cosine functions are very similar indeed to each other.
As you might guess, there's a way to describe this close similarity between the two functions mathematically. Look again at the curves and choose any value for q. The value of the sine function at that point q is the value of the cosine function a bit further back, for a smaller q. In fact this smaller value is exactly p/2 smaller. So sin(q)=cos(q- p/2). Or we could put it the other way and say cos(q)=sin(q+ p/2).
That's our second relationship between sine and cosine:
sin(q)=cos(q- p/2) or equivalently cos(q)=sin(q+ p/2).