I tried to learn this by myself and I understand the technic but not the 'why'. For example, I came across a heat transfer problem where:

T(x,y)=A * sin(n*PI*x/a)*e^(-n*PI*y/a)

with a boundary condition which was not used and it is: T(x,0)=f(x)

Now, if I want to express T(x,y) I guess I need to find A. This is simple, and I do not need fourier series theory for that, because I have the boundary condition T(x,0)=f(x). So, why to use fourier series here at all? Is it because we do not know which 'n' is to be used?