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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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There is no single answer. If you're using angles to represent the relative orientation of two lines, say, then $0$ is the same as $2\pi$. If your angle represents, on the other hand, how much you've twisted one end of a torsional spring, then $0$ is not the same as $2\pi$. And then there are situations where $0$ is not the same as $2\pi$ but is the same as $4\pi$... – Rahul Jan 17 '12 at 7:57
I think "!" should be omitted in the third sentence from above. And you can edit "...this belongs to N,R,C..." to be "this belongs to $\mathbb{N}$ , $\mathbb{R}$ ,$\mathbb{C}$ ... – Babak S. Jan 17 '12 at 8:55
When is one thing equal to another – Seamus Jan 17 '12 at 11:25
It's almost "ad-hoc". – Strin Jan 17 '12 at 11:51

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.

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