Why, in physics, isn't the law of sine used as often as the law of cosine?

Question

When dealing with vector quantities (like force) in physics, we often have to either decompose the vector into appropriate components or find the length of the opposite side of a non-right-angle triangle.

To give a more specific example, for those who know physics, I can mention the way the distance between a mass element and the point P in the proof of Newton's shell theorem (see the diagram below).

Back to the original point, in the process of expressing this distance we use the cosine law, not the sine law; and indeed, if I apply the sine law, the expression for $dF$ (force element) becomes a bit dirty and very hard to integrate. But I'm curious why that is. The sine law is much simpler than the cosine law, but why is this happening? Or am I wrong and is it possible to use the sine law in integration?

Answer 1

A purely heuristic argument perhaps runs as follows. Classical mechanics assumes Euclidean geometry to be the underlying geometry, where, given two vectors $u,v$, with an angle of $\theta$ between them, the cosine-law is expressed as follows

$$\|u\pm v\|^2=\|u\|^2\pm2\cos\theta\cdot \|u\|\cdot\|v\|+\|v\|^2$$

and since in many cases a quantity of interest is in fact $\|u\pm v\|$ ,(given $u,v$), the cosine rule seems to be more susceptible than the sine rule.

Answer 2

The sine law may appear simple, perhaps only deceptively so, at least for your example where $\theta$ is the only known angle. Because the sine law would require two angles, instead of just $\theta$ as in the cosine law.

If you eliminate the second angle in terms of $\theta$ in the sine law, you would end up with the same expression of the cosine law.