Confusion regarding eigenfunction expansions and Fourier series

Question

I am aware of the fact that a Fourier series is an example of an expansion of orthogonal functions. In particular, it can be shown that the sequence of eigenfunctions for the periodic Sturm-Liouville boundary value problem $y'' + y*\lambda = 0$; $y(0) = y(2\pi)$, $y'(0) = y'(2\pi)$ forms the orthogonal basis for the Fourier series: ${\frac{1}{2}, \cos(x), \sin(x), \cos(2x), \sin(2x)...}$. However, this is not an orthonormal basis. Thus when we are finding the coefficients for the Fourier expansion, we must divide by a scaling constant to account for this. Do the formulas for the Fourier coefficients account for this in some way?