Is there an explicit formula for the inverse of $\cot\left(\frac{x}{2}\right)\sqrt{1-\cos(x)}$?

Question

I apologize if this is trivial but I am stuck.

Given the bijective function $f:(0,2\pi) \to (-2,2)$ with $$ f(x)=\cot\left(\frac{x}{2}\right)\sqrt{1-\cos(x)} $$ where $\cot$ is the cotangent, how can I find an inverse $g:(-2,2)\to (0,2\pi)$? Is there an explicit formula?

score 5 · Accepted Answer · answered May 15 '13 at 11:29

5

Hint: Use this fact that $$\frac{1-\cos(x)}{2}=\sin^2(x/2)$$ and that if $x\in(0,2\pi), |\sin(x/2)|=\sin(x/2)$

answered May 15 '13 at 11:29

Mikasa

67,374

Spot on, my friend!! $\Large +1$ – amWhy May 16 '13 at 01:09
@amWhy: I am good and I am manipulating elements in a semigroup to generalize it. So hard Amy! wish me luck. – Mikasa May 16 '13 at 05:05
Good luck! Is this for a presentation? a paper? – amWhy May 16 '13 at 05:06
@amWhy: I missed your kind comment here, so sorry for myself :..( Yes for a paper. – Mikasa May 16 '13 at 15:36

Is there an explicit formula for the inverse of $\cot\left(\frac{x}{2}\right)\sqrt{1-\cos(x)}$?

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