the Fourier coefficient formulas
If a function really is a sum of sines and cosines, how do you find out HOW MUCH of each is in there? You cannot just stare at the function and read off the amplitudes. The Fourier coefficient formulas are the precise integral recipes that extract each amplitude one at a time, like a tuning fork that resonates with one frequency and ignores all the others.
For a function f of period 2L the formulas are a_0 = (1/L) integral over one period of f dx, a_n = (1/L) integral over one period of f(x) cos(n pi x / L) dx, and b_n = (1/L) integral over one period of f(x) sin(n pi x / L) dx. The trick that makes them work is orthogonality: multiply the whole series by, say, cos(m pi x / L) and integrate over a period. Every cross-term — cosine times a different cosine, or cosine times any sine — integrates to exactly zero. Only the term that matches survives, and it leaves behind a single coefficient times L. Divide by L and you have isolated a_m. The integral is literally projecting f onto one basis wave.
These formulas are the engine of every separation-of-variables computation. Once you have expanded the initial data of a PDE in eigenfunctions, the coefficients are precisely these integrals, and they tell you the amplitude of each separated mode. The same projection idea generalizes far beyond trig functions: with a weight function it gives the coefficients of a Bessel or Legendre expansion, and abstractly it is just taking inner products against an orthogonal basis.
To find b_3 for a period-2pi function f, compute b_3 = (1/pi) integral from -pi to pi of f(x) sin(3x) dx. Multiplying f's series by sin(3x) and integrating kills every term except the one containing sin(3x): orthogonality gives integral of sin(3x) sin(nx) = pi if n = 3 and 0 otherwise. So the integral reads off exactly the amount of sin(3x) hiding in f, and nothing else.
Each coefficient is an inner product of f with one basis wave; orthogonality is what lets you read them off one by one.
Normalization conventions differ between books: factors of 1/L vs 2/L vs 1/(2L), and whether the constant term is written a_0/2 or a_0. Always check which convention is in force, or your constant term will be off by a factor of two.