This question already has an answer here:
When we are introduced to waves in school, we are often presented with a picture of a sinusoid (or a cosinusoid).
Sinusoids can represent the way many physics phenomena behave, still....
Why are sinusoids so common?
When I first started to be more interested in waves, I was confused.
According to me a triangular wave, or a square wave seemed to be much more "intuitive". At least "easier". Why aren't we choosing them?
Mathematic gives us a little clue, but it doesn't show the full picture.
In fact, they form an orthonormal basis of the vector space of functions on $[0,2\pi]$. (More or less, every function in an interval can be written as a linear combination of them).
Moreover, sines and cosines are eigenfunctions (eigenvectors) of the second-derivative operator (eigenvalue $=-1$).
So sines and cosines are a solution of the [ $x''=-kx$ ] differential equation that is produced by a simple harmonic oscillator, which can be basically anything that obeys $F=-kx$.
But how do we glue all these pieces together to understand why are sinusoids such a significant phenomenon of our world?
Intuitively, I can guess that given the "apparent" analog nature of our world, physic phenomena presenting a back and forth pattern must go through a smooth transition as they reverse their cycle, and this may be why sinusoids are perfect in embodying this job. Though, my road to get a full picture seems still very long.