Why is periodic motion always represented as a sine function

Question

Why was periodic motion started to be represented as a sine function at the beginning? Is it because it is perhaps the simplest periodic function? Is there certainly any reason behind this representation?

Answer 1

Periodic functions can be also represented by cosine functions, linear combinations of sine and cosine, and more generally by linear combinations of exponential functions $e^{int}$ which represents a circular motion with fixed angular speed.

Of course, they can be also represented differently, like for example zig-zag functions used in ancient Babylon. The main mathematical advantage of exponentials come from the fundamental fact that they are eigenfunctions of the shift operator: $$e^{in(t+a)}=e^{ina}e^{int}.$$ This fundamental property implies all other useful properties.

The idea that "every (celestial) motion can be represented in terms of uniform circular motions" was promoted by Greek philosophers in 4th century BC, and it was implemented in practice by Ptolemy and later astronomers.

Added. In my earlier answer I said that Fourier was not influenced by astronomy. He was solving different problems, related to heat equation. But the prominent role of sine and cosine in his work is explained by the SAME fundamental reason: they are eigenfunctions of the shift operator. And this fundamental property explains their omnipresence in mathematical physics and other areas of mathematics.

Answer 2

All periodic motions can be represented by (finite or infinite) sums of sines and cosines (Fourier).

You could have started with other basis functions. Like zigzag functions with different frequencies. The fact is though that most waves in Nature are caused by vibrations obeying Hooke's law, which has sine or cosine solutions.

There are other kinds of periodic processes which obey different (non-linear) equations. These processes can't be approximated by a superposition of sines or cosines ( no superposition). Like the waveforms resembling a curved mountain with a sharp peak. I forgot their name.

I'm not sure if this is a history question but it's a fact that physics stayed in the linear realm a long time. Non-linear phenomena are much more interesting. But much more complex. If mechanics had started out non-linearly then maybe today you would have learned about non-linear signals already at high school. Although... Computers were not available in the time of Newton and Huygens. And you need these to do the intensive numerical calculations going along with non-linearity. To do this by hand was even for Newton too much.(sorry for eventual spelling mistakes; I type on a phone...).

By the way, the non-linear "mountain" waves are called peakons.

Already in the 17th century Hooke invented his law. It is a linear law (pull a spring twice as far and the opposing spring force gets twice that big). Solutions to this law are sines and cosines. They can both involve time and distance so they can represent waves, which were thought to be linear phenomena (which they are not in reality but the appriximation showed to hold very well).