For an orthonormal set, one also gets another answer to your question: how can you see how much such a set spans?
Theorem: If S is an orthonormal (or even orthogonal) set in a Hilbert space V (here, L^2 functions on R/Z), then put S^\perp = {v in V : v is orthogonal to every s in S}. Then the span of S is (S^\perp)^\perp. (The containment of the span in the double-orthocomplement is formal; the other direction requires a supplemental theorem on the geometry of Hilbert spaces.)
With this in mind, we see that S is a topological basis (spans everything) when S^\perp = 0; so, to show that the sine and cosine functions span, it suffices to show that nothing (non-0) is orthogonal to all of them. This still requires computation, but at least it's easier to imagine doing this than somehow finding a Fourier series for any L^2 function.
Theorem: If S is an orthonormal (or even orthogonal) set in a Hilbert space V (here, L^2 functions on R/Z), then put S^\perp = {v in V : v is orthogonal to every s in S}. Then the span of S is (S^\perp)^\perp. (The containment of the span in the double-orthocomplement is formal; the other direction requires a supplemental theorem on the geometry of Hilbert spaces.)
With this in mind, we see that S is a topological basis (spans everything) when S^\perp = 0; so, to show that the sine and cosine functions span, it suffices to show that nothing (non-0) is orthogonal to all of them. This still requires computation, but at least it's easier to imagine doing this than somehow finding a Fourier series for any L^2 function.