We have an angle $\alpha$, then we have $\alpha +2k\pi$. How are these angles considered between themselves? When we apply trigonometric functions they have the same value but that is not always true like considering them as real numbers. $(\alpha +2k\pi)\sin(\alpha +2k\pi) \neq \alpha \sin\alpha$.
So maybe we should define where $\alpha$ belongs, its algebraic structure. My knowledge is limited but I usually see, this belongs to $\mathbb N, \mathbb R,\mathbb C \dots$ Is there a similar thing for angles, when people define a problem, an expression, formula, do they say "It belongs to Angles"? Or are angles always considered as complex numbers and then we are obliged to use and differentiate between $\alpha$ and $\alpha +2k\pi$?