Sines and cosines are the best choice here because they're the "elementary periodic function" for a given frequency: the sum of two (possibly scaled and shifted

Sines and cosines are the best choice here because they're the "elementary periodic function" for a given frequency: the sum of two (possibly scaled and shifted) sinusoids at some frequency is also a (scaled and shifted) sinusoid at the same frequency. For other shapes of periodic function it's not true: for example, if you sum two triangle waves of the same frequency but offset from each other, you'll get a shape that's more complex than a triangle wave. Same for two square waves etc. So the most natural way to decompose a signal is into a sum of sinusoids rather than triangles or squares etc.