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I want to find the general formula to calculate the slope of the bold line in the picture when we put one side of the regular $n$-gon upside. I have found $3$ data below, even though I haven't verified the third data. Note that their sides are identical in length. I suppose, there's something connected with tangent. I.e.:

Let $m$ be the slope. I have found that

$$m_{3} = \tan\left(\frac{\pi}{2}\left(\frac13\right)\right)= \frac{\sqrt3}{3}$$

$$m_{4} =\tan\left(\frac{\pi}{2}\left(\frac12\right)\right)= 1$$

$$m_{5} =\tan\left(\frac{\pi}{2}\left(\frac23\right)\right)= \sqrt3$$

And note that:

$$\frac13 < \frac12 < \frac23 < \cdots < 1$$

I give the supremum $1$ because we know $\tan{90^°}$ is undefined. The reason behind it, I believe, because the slope is getting bigger and bigger and we also know that straight vertical line has infinity slope if I understand correctly. Only if I could find the pattern, would I be able to find the formula.

user516076
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1 Answers1

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The angle is $\dfrac\pi2-\dfrac12\cdot\dfrac{2\pi}n$. The slope is $$ \tan\big(\dfrac\pi2-\frac12\cdot\dfrac{2\pi}n\big)=\cot\big(\dfrac{ \pi}n\big) $$ For large $n$ (small argument): $$ \cot\big(\dfrac{ \pi}n\big)\to\frac n\pi-\frac\pi{3n}-\frac{\pi^3}{45n^3}+... $$

am301
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  • Thank you so much! Is it a lucky guess or there's a story behind it? If you have spare time, could you elaborate, please? – user516076 Nov 21 '21 at 06:24
  • It is not a guess but just a matter of defining the angle right. BTW I think you have an error in your caculation of $m_5$ it should be $\pi/2-(1/2)2\pi/5=(\pi/2)(3/5)$ – am301 Nov 21 '21 at 06:31