For so long I've taken COS 1° as 0.540302305 in Radians and I can get that if I say to my calculator SIN(PI/2-1°), though I'm stumbling what SIN does to that result and I probably did this stuff heaps of times at school and asked myself how's this going to help me in Life.

It's virtually the same formula to get SIN except you substitute COS for SIN in that problem above.

This is the site I was getting my information at but is there anyway to explain it to an idiot?

Honestly, that webpage does a much better job of explaining things than I ever could! Just carry on reading that page, but certainly the section "Series Definitions" is probably the information you're after.

Personally, I always find it easier to think of this stuff in terms of complex numbers, although it will seem very complicated at first, especially if the above seems hard. But essentially, you can think of a point on a circle, instead of as being x=r*cos(a), y=r*sin(a), as being p=r*(cos(a)+i*sin(a)). The interesting thing about complex numbers is that i*i = -1. If you look back at the series definitions, you'll see that sin and cos seem to be alternating terms in the same equation. The secret is that if you multiply the sin ones by i and then add them together, you'll see that the equation is sum( (xi)^n/n! ). The special way of writing p is e^(xi) but that requires quite a lot of maths to prove and also it doesn't really help understanding at this point, so maybe just accept that as true for now...

Also, if you remember that cos(x)=sin(x + pi/2), you can think of that in the complex plane by looking at the diagram, and think what adding pi/2 to the angle means... it's simply a rotation (generally drawn anticlockwise) of the point. You can also think of it as multiplying by i, which is the same operation, and likewise with the combined equation above (the cos(x)+isin(x) added together) and write it out, you can see how that effectively just shift each term along one and so how the cos turns into a sin.