If i have a series of the form ( say fourier or anything ) , For example lets consider $$\sum_{k\in \mathbb Z} \exp\left( \frac {-ik\pi}{L}\right) a_k(t) ,$$ is it always possible to split it down to something like $$\sum_{k=1}^\infty b_k(t) \sin\left( \frac{k\pi x}{L}\right)+c_k(t) \cos\left(\frac {k\pi x}{L}\right)?$$
What I am missing here is I think the splitting of coefficients.
It may be a silly question but i am still missing.
Any help will be appreciated.