It's because the eigenvectors of the collection of all time shift operators are the exponential functions, and if you want to stick with real functions, then sin(ax) and cos(ax) together form a basis for the sum of the ai and -ai eigenspaces. (In the non-periodic case, you also have to deal with e^(cx) and products of this with sin and cos for all the "transients"). If you are doing something that is time-invariant, that means it commutes with time shift operators, and that means it has the same eigenvectors. Importantly, this implies such an operator can be analyzed (more or less) by how it scales these eigenvectors.