If cosine is just sine rotated by 90°, then we really just use sines. Except we write cosine instead of a sine with a 90° rotation because it looks less confusing :)
The real goal is to arrive at the exponential representation, where the sines and cosines are absorbed into the exponential functions, which is later needed to write down the fourier integrals.
Your question boils down to preference. The two things you name are, in fact, equal. Math people just choose to write cos(x) instead of sin(x+pi/2)
If cosine is just sine rotated by 90°, then we really just use sines. Except we write cosine instead of a sine with a 90° rotation because it looks less confusing :)
The real goal is to arrive at the exponential representation, where the sines and cosines are absorbed into the exponential functions, which is later needed to write down the fourier integrals.
Your question boils down to preference. The two things you name are, in fact, equal. Math people just choose to write cos(x) instead of sin(x+pi/2)