Right, you need both sine and cosine parts, hence e^ix, in order to fully describe the function. Unless the function is completely odd (or even), in which case the transform really is equivalent to `x[n] * sin(2pik*x/N)` (or cos). To see why imagine analyzing a function W that is just a plain cosine wave (any frequency and amplitude). If we only use the sine part of the Fourier transform, F(W) is indistinguishable from F(-W). In fact both are zero everywhere.

Transforms that use only sines or cosines (like the DCT) provide a complete basis by increasing in frequency by only a half cycle (pi) rather than by integer cycles (2pi). Essentially trading half a transform of sines and half a transform of cosines for one transform of half-cosines (or sines).