[NOTE : Whoops, sorry for that little dot below the a_1 images]

1) Take a function . We will multiply both sides of the equation with sin(x), the result being :

.

2) To find the coeffiencts a(1), a(2) – a(n), we take an integral of f(x) sin (x) from 0 to pi.

We can now find the value of , being

3) We found the formula of by multiplying the original function f(x) by sin(x) (). Now, to find the value of , instead of multiplying it by sin(x), we multiply it by sin(2x) – . Now, we repeat step 2, integrating .

If you go through proof by induction, you can proof the general result, although I won’t go into that now –

We have now found a general formula to find any coeffiecients given the function f(x). In the next section, we will discuss the use of this general formula.

Oh nO!!! I discovered a horrible mistake. Step 2)

int(fx sin x) = int (a_1 sin^2x) + int(fx sinx sin2x) …

should really be

int(fxsinx) = int(a_1 sin^2 x) + int (a_2 sinx sin2x) + int(a_3 sinx sin3x)

(the f(x) should be a_2, a_3, and so on. Sorry.