I am told that Fourier showed that we can represent an arbitrary continuous function, $f(x)$, as a convergent series in the elementary trigonometric functions
$$f(x) = \sum_{k = 0}^\infty a_k \cos(kx) + b_k \sin(kx)$$
Also, suppose that $\{\phi_n(x)\}^\infty_{n = 0}$ is a set of orthogonal functions with respect to a weight function $w(x)$ on the interval $(a, b)$. And let $f(x)$ be an arbitrary function defined on $(a, b)$. Then the generalised Fourier series is
$$f(x) = \sum_{k = 0}^\infty c_k \phi_k (x)$$
I have the following questions relating to this:
How does a Fourier $\sin$/$\cos$ series arise from a "normal" Fourier series $f(x) = \sum_{k = 0}^\infty a_k \cos(kx) + b_k \sin(kx)$?
How does this relate to the generalised Fourier series $f(x) = \sum_{k = 0}^\infty c_k \phi_k (x)$?
I would greatly appreciate clarification on this.
EDIT: When I say Fourier $\sin$/$\cos$ Series, I'm referring to what is known as "Fourier sine series" and "Fourier cosine series".