Many natural systems are well described by sinusoidal motion.
By Fourier theory, any waveform can be represented by a summation of a (possibly infinite) number of sinusoids, each with a particular amplitude and phase.
Thus, we can view sinusoids as fundamental ``building blocks'' for more complicated signals (as we saw with additive synthesis).
Sinusoids are relatively easy to represent and manipulate mathematically.
Because sinusoids are periodic functions (repeating every radians), they naturally represent periodic signals. On the other hand, one should expect that any attempt to decompose a non-periodic signal into sinusoidal components will have limitations.