Why are the laws of sines/cosines "laws" and not "theorems"?

Question

So in logic we have every line of a proof being either an axiom or a theorem -- but then why do we have concepts like the "law of sines" and the "law of cosines"? Are these technically "theorems" as well? How is "law" being used here? Is there a mathematical reason? A historical one?

Answer 1

This is a law because no other law can prove this. Though using basic mathematics can prove this. Like pythagoras theorem is a special case of cosine law hence it is called a theorem

Answer 2

One could be forgiven for thinking the terminology is a mess. There are other examples of laws/rules in mathematics, some trigonometric, but not in general. The term "identity" is also common (in fact, some of those links came from here). I could find nothing anywhere indicating when to use each term. However, based on my experience, the way people use the terms in the names of named results seems to go as thus (no doubt with plenty of exceptions):

• Any result of the form $f(x)=g(x)$ with the same variables $x$ on each side tends to be called an identity (whereas $f(x)\ne g(x)$ would be an inequation, and $f(x)\ge g(x)$ and $f(x)>g(x)$ would be inequalities), unless it governs the arithmetical operations on such variables, e.g. $ab=ba$, in which case "law" is typically used;
• Any equally simple to use result for calculations, which may use different variables on each side, is called a law or rule, sometimes interchangeably (e.g. $a=2R\sin A$ has been called both the law of sines and the sine rule, although for some reason you only seem to see "law" in trigonometry, e.g. no-one would refer to Cramer's law);
• A result you can use procedurally to verify a claim tends to be called a criterion;
• Other results tend to be called theorems: in particular, "for all $x$ there exists a $y$ such that..." would be a theorem.

Technically, all of these are theorems, but even if people acknowledge them as such, the name of the result might not reflect that.