This may seem backwards since a fourier series isn't typically used this way but I'm trying to prove whether or not the sum of sin and cos waves could produce a sin wave with a wave length that is not in any of the summed waves.
I don't intend the use of the fourier series as a restriction. It just seemed an obvious place to start thinking about this problem.
The restriction is, make a sin of finite wavelength L by summing any sin's and cos's so long as they do not have that same wavelength L.
If you are working with a finite interval, you have available to you the sine and cosine waves with frequencies that are multiples of $\frac {2\pi}L$ These waves are all orthogonal, so you cannot approximate any one as a sum of the others.