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?

  • 1
    $\begingroup$ Possible duplicate: math.stackexchange.com/questions/2476/… $\endgroup$
    – lhf
    Mar 15, 2011 at 1:10
  • $\begingroup$ See also mathworld.wolfram.com/NivensTheorem.html $\endgroup$
    – lhf
    Mar 15, 2011 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 $n\equiv 0\pmod 3$. $\endgroup$ Mar 15, 2011 at 1:14
  • $\begingroup$ @Joseph: $\sin \frac{\pi}{n}$ and $\cos \frac{\pi}{n}$ (I assume this is what you meant) are always algebraic. $\endgroup$ Mar 15, 2011 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, 2011 at 1:48

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.