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If we have a semicircle of diameter AB (with length 1) and center O, then we can find the approximate value of the chord AC, by the following formula,where $x$ represents the value of the angle AOC

$\sqrt\frac{1}{\left(\sqrt[90]\frac{10{\pi}}{\sqrt{50}×4}^{(2x-90)}×\left(\frac{90}{x}-1\right)×\sqrt{ 2}+1\right)^2+1}$

For example if the value of $x=60$, we will get 0.4999469.. So , my formila gives an approximate value of the chord. There is formula similar to mine, in the literature?

Srbin
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  • You can find the exact value through some trigonometry. – Vishu Mar 15 '21 at 13:36
  • $|AC| = 2r\sin(\frac x2)$. But wow, that's some function you've got there :O How in world did you come up with that? – Milten Mar 15 '21 at 13:53
  • In any case, your formula just looks like something that maybe happens to aproximate the real answer on some range. Unless you can say something surprising about it, I don't think it's significant at all. – Milten Mar 15 '21 at 14:02
  • If there were anything like this in the literature, I imagine it would have a greatly-simplified form; for instance: $$\frac{1}{\sqrt{1+\left(1+\dfrac{90-x}{x}\cdot\dfrac{\pi^{(x-45)/45}}{2^{(x-60)/30}}\right)^2}}$$ Or, perhaps better, defining $t := x/90$, $$\frac1{\sqrt{1+\left(1+\dfrac{1-t}{t}\cdot\dfrac{\pi^{2t-1}}{2^{3t-2}}\right)^2}}$$ – Blue Mar 15 '21 at 14:52

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