The "true" Fourier transform is to write the function as an infinite sum of e^(2 pi i n x) with complex coefficients and n an integer. (This is the "true" Four

The "true" Fourier transform is to write the function as an infinite sum of e^(2 pi i n x) with complex coefficients and n an integer. (This is the "true" Fourier transform because of a connection between periodic functions and circles, and because the representation theory of the circle group says that these are the so-called irreducible representations. This is extremely abridged -- the point is just to mention relevant words.) Due to Euler's formula, there is a correspondence between a linear combination of e^(2 pi i n x) and e^(-2 pi i n x) and a linear combination of cos(2 pi n x) and sin(2 pi n x). In other words, both pairs of functions span the same subspace (with complex coefficients). We need all of the exponentials to represent a periodic function, with positive and negative n, and so we need both cos and sin. If you leave out cos, you can only represent odd functions, and leaving out sin, even functions. But: if your question is just why not just use sin(x) and sin(x-pi/2) instead of sin(x) and cos(x), well, it's the same thing. Just because you write cos(x) like sin(x-pi/2) doesn't mean it's not still cos(x)! (Fun fact: cosine means "complementary sine," the sine of the complementary angle in the triangle.)