Fourier coefficients and orthogonal projection

Orthogonality of cosines. For $m\neq n$:
$$\langle\cos mt,\cos nt\rangle=\frac1\pi\int_{-\pi}^\pi\cos mt\cos nt\,dt=\frac1{2\pi}\int_{-\pi}^\pi[\cos(m-n)t+\cos(m+n)t]\,dt=0.$$
For $m=n$, it equals $1$.

Orthogonality of sines. Similarly, $\langle\sin mt,\sin nt\rangle=\delta_{mn}$.

Orthogonality between cosine and sine. $\langle\cos mt,\sin nt\rangle=0$ because the integrand is odd.

Orthogonality with constant. $\langle 1,\cos mt\rangle=0$ and $\langle 1,\sin nt\rangle=0$ (integral of cosine/sine over a period).

Conclusion. These functions form an orthogonal family. The Fourier coefficients $a_n=\langle f,\cos nt\rangle$, $b_n=\langle f,\sin nt\rangle$ are the projections of $f$ onto these orthogonal functions.