I have a hypothesis about regular polygons, but in order to prove or disprove it I need a way to determine whether an expression is rational. Once I boil down my expression the only part that could be irrational is:

$$S_N = \cot \frac{\pi}{N} \text{ for } N\in ℕ_1 ∖ \left\{1, 2, 4\right\}$$

Is there at least one such $N$ for which $S_N$ is rational? Can it be proven that $S_N$ is never rational for any such $N$? How would I go about proving one or the other?

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    $\begingroup$ Possible duplicate: math.stackexchange.com/questions/2476/… $\endgroup$ – lhf Mar 15 '11 at 1:10
  • $\begingroup$ See also mathworld.wolfram.com/NivensTheorem.html $\endgroup$ – lhf Mar 15 '11 at 1:11
  • $\begingroup$ I think (but this could be a dubious claim) that there is a theorem saying that $\sin(\pi n)$ and $\cos(\pi n)$ are algebraic numbers iff $n\equiv 0\pmod 3$. $\endgroup$ – Chris Brooks Mar 15 '11 at 1:14
  • $\begingroup$ @Joseph: $\sin \frac{\pi}{n}$ and $\cos \frac{\pi}{n}$ (I assume this is what you meant) are always algebraic. $\endgroup$ – Qiaochu Yuan Mar 15 '11 at 1:26
  • $\begingroup$ I don't think this is a duplicate, though it is certainly related. Please see my comment to Ross Millikan's question below. $\endgroup$ – kojiro Mar 15 '11 at 1:48

A simple, complete proof can be found in Olmsted, J. M. H., Rational Values of Trigonometric Functions, Amer. Math. Monthly 52 (1945), no. 9, 507–508.


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