Let \(f(x)\) be a piecewise function continuous on \([0,L].\) The Fourier cosine series of \(f(x)\) on \([0,L]\) is
$$
\frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos \frac{n \pi x}{L},
$$
where
$$
a_n = \frac{2}{L} \int_0^L f(x) \cos \frac{n \pi x}{L} \; dx, \quad n = 0,1, \ldots
$$
The Fourier sine series of \(f(x)\) on \([0,L]\) is
$$
\sum_{n=1}^\infty b_n \sin \frac{n \pi x}{L},
$$
where
$$
b_n = \frac{2}{L} \int_0^L f(x) \sin \frac{n \pi x}{L} \; dx, \quad n = 1,2, \ldots
$$