How can we acquire the coefficients of the trigonometric series?

Actually, the trigonometric series and Fourier series are the same, right? If not, please tell me how they differ. Anyway, I want to find the coefficient of this trigonometetric series(or partial sum).

$$f(x)=a_0 + \sum\limits_{n=1}^N a_n cos(nx) + b_n sin(nx)$$

Answer

You need to project your function on the basis vectors using

$$a_{0}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)dx$$

$$a_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)cos(nx)dx, \:n>0$$

$$b_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)sin(nx)dx, \:n>0$$