If I understand Fourier series correctly. The objective is to model a periodic function as a linear combination of sinusoidal functions with different amplitudes, frequencies, etc. Since the collections $\{\frac{\cos(nx)}{\pi}\}_{n=0}^{\infty}$ are orthonormal over in the $L^2([-\pi,\pi])$ norm then essentially I can find the coordinates of $f(x)$ in this new basis by taking the inner product
$$C_n=<f(x),\frac{\cos(nx)}{\pi}>=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)dx$$ to form the series:
$$\sum_{n=0}^{\infty}C_n\cos(nx)$$
Similarly we can find the projection onto the orthonormal collection $\{\frac{\sin(nx)}{\pi}\}_{n=1}^{\infty}$
To get
$$f(x)=\sum_{n=0}^{\infty}C_n\cos(nx)+\sum_{n=1}^{\infty}B_n\sin(nx)$$
If I am mistaken in any way please correct me. My question is why does the projection have to be onto both $\sin(nx)$ and $\cos(nx)$? Does this have to do with the fact that $\sin(nx)$ is an odd function and $\cos(nx)$ is an even function therefore their linear combinations can replicate more complicated periodic functions. That is $\cos(nx)$ is well equipped to model even periodic functions and $\sin(nx)$ is well equipped to model odd periodic functions, but any periodic function can be uniquely decomposed into odd and even parts therefore we need both to model any periodic function. Finally, when the projection is done using exponential form that is the collection $\{e^{-inx}\}$ why does the fourier series have to be a two tail that is
$$\sum_{-\infty}^{\infty}A_ne^{-inx}$$
Best Answer
Any periodic function $f$ can be uniquely decomposed into an odd part $f_o$ and an even part $f_e$ as below: $$ f_o(x)=(f(x)-f(-x))/2, $$ and $$ f_e(x)=(f(x)+f(-x))/2 $$ Not only that, a purely odd function cannot be represented as a linear combination of even functions, and vice versa. So just sines or just cosines are not enough to represent all periodic functions.
As for the other question, think of the formulae $$ \cos(x)=\frac{e^{ix}+e^{-ix}}{2} $$ and $$ \sin(x)=\frac{e^{ix}-e^{-ix}}{2i} $$ and it all becomes clear.