I think it's my ignorance of not accepting that I can't do it, because

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I think it's my ignorance of not accepting that I can't do it, because I'm almost certain that it cannot be done, yet I'm still asking.

Anyway, can this equation be solved for m?

$$\frac{\pi}{n+1}=cos^{-1}\left(\frac{1-m}{\sqrt{1+m^2}}\right)+\frac{(m-1)\sqrt{2m}}{1+m^2}$$

If it cannot be done (which I'm almost certain will be the case), for a given n, how can I still find the value for m? Be it a numerical value if it must...

The equation can be simplified to
$$\frac{2\pi}{n+1}=\alpha-\sin\alpha$$
where
$$\cos\frac{\alpha}{2}=\sqrt{\frac{2m}{1+m^2}}$$
and then has to be solved numerically.

Please check if I didn't do a mistake.

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