2

What I'm claiming is that I have devised a proof of this statement which says that it is true. The proof is a little long and involves trigonometry and basics of number theory, so I feel lazy to include it here. I want to know whether such a result exists or not from before. If there exists one, then please refer me to the very website(s) containing the existing proofs.

I've used the identity, (1+x2)n = (nC0 – nC2x2 + nC4x4 - ...)2 + (nC1x – nC3x3 + nC5x5 - …)2 to show that tan(pπ/q) with p≠0 and odd q is never rational.

Next, I've analyzed the zeroes of the polynomial (nC0 – nC2x2 + nC4x4 - ...) to show that tan (pπ/q) with odd p and even q is never rational except for (p/q)=±0.25.

...and still sorry for being lazy.

Indrayudh Roy
  • 2,256
  • 13
  • 22
  • I bet you, lazy guys would not get help here! – Mahdi Khosravi Jul 26 '13 at 11:18
  • To the serial downvoters: notice that this question clearly is a reference request (although the OP didn't tag it as such), so there's no problem with the OP being lazy. – Git Gud Jul 26 '13 at 11:19
  • To Shobhit, Babak S. and Michael Albanese (who where the people involved this edit). Can any of you explain to me the comma after then in If there exists one then, please refer...? – Git Gud Jul 26 '13 at 11:28
  • 1
    Granted, monsieur, you have proved the statement. Some one else might have proved this in a different way than what you have done. Only if you show us your proof can we check whether it has already been done! – Torsten Hĕrculĕ Cärlemän Jul 26 '13 at 11:36
  • It is well-known exactly which rational $x$ result in rational values of $\tan\pi x$, $\sin\pi x$, and so on. You can probably find the details in Niven's book on irrational numbers. – Gerry Myerson Jul 26 '13 at 13:18
  • Essentially answered at this question: http://math.stackexchange.com/questions/79861/arctan2-a-rational-multiple-of-pi – ShreevatsaR Jul 26 '13 at 13:27

1 Answers1

0

It's proved at http://www.oberlin.edu/faculty/jcalcut/tanpap.pdf (J. S. Calcut, Rationality and the tangent function).

Gerry Myerson
  • 179,216