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In the case of a sphere, the azimuth angle is $$2\pi$$ periodic in that if you add $$2\pi$$ to any azimuth angle you get the same direction. Similarly, the polar angle is $$\pi$$ periodic.

In stating this we have to assume that the domain of the azimuth angles are 360 degrees and the polar angle is 180 degrees.

What is the proper and precise way to announce that it is acceptable for you to, in the context of a proof, add $$2\pi$$ to an azimuth angle or $$\pi$$ to a periodic angle without affecting any equalities?

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    The polar angle isn't $\pi$-periodic. Polar angle of $1$ and polar angle of $\pi+1$ doesn't give you the same point at all. – Arthur Jan 10 '18 at 20:11
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    Aside from the issue Arthur mentions, if a quantity truly is $x$-periodic for some $x \in \mathbb{R}$, there's nothing wrong with just announcing in the proof, "Such and such quantity is $x$-periodic, so any multiple of $x$ can be added to it without affecting its value. Thus . . ." Then proceed with what you wanted to do involving the periodic quantity. – wgrenard Jan 10 '18 at 20:21
  • I don't think anything needs to be said. It is understand angles are period mod $2\pi$ and as Arthur points out the polar angle is not $\pi$-periodic. It's $2\pi$-periodic. It's that angles with polar angle greater than $\pi$ are redundant. – fleablood Jan 10 '18 at 20:25

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