Fourier series represent a wave as a summation of harmonically-related sines and cosines.
I am wondering if it is possible to break down a natural sound as harmonically related sawtooth/square/triangular waves? If it is, why didn't mathematicians use these waveforms rather than sines and cosines?
The trigonometric functions are related through the exponentials through Euler's formula, $$e^{inx} = \cos (nx) + i\sin(nx).$$ A great advantage of the exponentials is that they are orthogonal in $L^2[0,2\pi]$. That is, for any $n,m\in \mathbb{Z}$ $$\left<e^{inx},e^{imx}\right> = \int_0^{2\pi} e^{inx} e^{-imx}dx =\begin{cases}0&m\neq n, \\ 1& m=n. \end{cases} $$ To make a long story short, it follows that we can create an orthonormal basis $\{e^{inx}\}_{n=-\infty}^\infty$ for $L^2[0,2\pi]$.
Thus, if you express a function as a combination of trigonometric functions, you are basically expressing it in terms of the orthonormal basis $\{e^{inx}\}_{n=-\infty}^\infty.$ If you've learned linear algebra, you would know the advantages of breaking down a function into "orthogonal components." Basically we are projecting a function into its different 'frequency levels.'
Of course, it is possible to use another basis to break a function down, with for example sawtooth functions, but it should be checked whether the basis is orthonormal and whether it really is, at least, dense in $L^2$.