I believe the reason is due to cosine being even, i.e. cos(-x) = cos(x), and sine being odd, sin(-x) = -sin(x). The sum of even functions is still even, and the

I believe the reason is due to cosine being even, i.e. cos(-x) = cos(x), and sine being odd, sin(-x) = -sin(x). The sum of even functions is still even, and the sum of odd functions is still odd. If you have a function that is symmetric about the origin, say f(x) = x^2, then you can approximate it with only cosines, but in general functions are not odd or even so you need to use a mix of sines and cosines.