[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.