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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 $\alpha +2k\pi$?

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    It's almost "ad-hoc".2012-01-17

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As a partial answer, I would say that the real numbers $\alpha$ and $\alpha+2\pi$ are equivalent angles in the sense of magnitudes of rotation (the magnitude of rotation is the arc length along a unit circle centered at the center of rotation traversed by a point 1 unit from the center of rotation) because, about the same center, the images of the plane under a rotation of $\alpha$ and a rotation of $\alpha+2\pi$ are identical.