Understanding trig interval

Question

I have kind of a random question I'm hoping someone could help me with.

So I was thinking about the interval $[-\pi, \pi]$ for a trig functions. Isn't this is the same interval as $[0, 2\pi]?$ The reason why I say that (and maybe this is where my confusion is) is because couldn't $[-\pi, \pi]$ be split as $[-\pi, 0] \cup [0, \pi],$ which when drawn the angle of rotation hits all the vital points on the unit circle just like $[0, 2\pi]?$

Thanks in advance!

Answer 1

I feel it would depend on the specific trig function only if you include hyperbolic.