This paper shows that if we consider odd functions on $(-\pi,\pi)$ in $L_2$, then the only $2\pi$-periodic function $f$ for which $f(nx)$ is a complete orthogonal system is the sine function.
I'll loosely interpret this as saying that the conventional periodic harmonic analysis is only possible with sines, which is very non-intuitive. Why can't I use the square waveform, or the triangle waveform, or some smooth waveform (similar to sine but different) as a basis for harmonic decomposition?
Sure, that paper gives a proof. But their proof is a proof by exclusion. They don't use any properties of sines except that sines form a complete system. They essentially prove that there can only be one complete system of this form, and since we already know that sines are complete, that precludes other such systems from existence.
But that proof doesn't explain why is the sine so special; what's the unique property that only it has? Put another way, if I were Fourier, how would I know to consider sines, except by trial and error? Is it possible to derive this waveform from the fact that $f(nx)$ is orthogonal and complete in the space of odd functions on $(-\pi,\pi)$?
Update. Thanks for the historic background, but this question is more concerned with math than history. Namely, how one can "compute" $\sin x$ as the only $2\pi$-periodic $f$ for which $f(nx)$ is orthogonal and complete in the space of odd $L_2$ functions on $(-\pi,\pi)$. Not how one might guess the answer, or how one particular Jean-Baptiste Joseph Fourier guessed it.
Remember that orthogonal means nothing if you do not specify the scalar product (norm) you're using to define it. There are other sets of functions that are orthogonal and complete but they are so under different norms. Sine and cosine are only special in a Hilbert Space where the norm is the one defined by Fourier.