Just because it's the same number doesn't mean it's the same relationship.

Any definition that relies on trig (sometimes disguised as trig expansion) assumes the existence of a flat Euclidean space - which is a convenient mathematical abstraction in real-world terms, but not required in the general case.

Most everyday physics assumes a flat space, so it's not surprising certain constants fall out of this.

There are more complex and interesting definitions which rely on relationships defined by group theory. I don't know enough about those to say if they're just more complex ways to operate in a flat space, or more complex ways to disguise simple trig expansions, or if they're completely independent, or if something else is going on.