Complex signal sine reconstruction
A sum of sines/cosines (with frequencies that are integer multiples of a fundamental frequency) will always result in a periodic function (that's what Fourier series are about). But you can of course approximate any function with finite support by choosing the length of the function's support as your period (if you don't mind that the approximation will be a periodic continuation of the given function).

Note that the approximation by a Fourier series is a least squares approximation, not a point-wise approximation. So if the desired function has discontinuities there will always remain some error of fixed magnitude, no matter how many sinusoids you use for the approximation. This oscillatory behavior near discontinuities is called Gibbs phenomenon.