35
$\begingroup$

A while ago I came across the following identity in an online math forum (of which I don't remember the name): $$\tan\left(\frac{\pi}{11}\right)+4\sin\left(\frac{3\pi}{11}\right)=\sqrt{11}.$$

It is not hard to give a proof by rewriting everything in terms of $\exp(i\pi/11)$ and applying a sequence of manipulations. I am wondering where this identity is coming from. Can somebody think of a geometric interpretation? Of an algebraic one?

Edit: Here's an example of what I mean by an algebraic interpretation: The identity $$\sin\left(\frac{\pi}{7}\right)\cdot\sin\left(\frac{2\pi}{7}\right)\cdot\sin\left(\frac{3\pi}{7}\right)=\frac{\sqrt{7}}{8}$$ expresses the fact that for the Chebyshev polynomial $$T_7(x)=x(64x^6-112x^4+56x^2-7)$$ the product of the roots $\displaystyle \sin\left(\frac{k\pi}{7}\right)$, $1\leq k<7$, of the second factor is equal to the normalized constant term $\displaystyle \frac{7}{64}$.

  • 2
    You also have the similar but less interesting $\tan\left(\frac{\pi}{3}\right)+4\sin\left(\frac{3\pi}{3}\right)=\sqrt{3}$.2011-04-24
  • 5
    A proof of a similar identity can be found here: http://math.stackexchange.com/questions/11246/how-to-prove-that-tan3-pi-11-4-sin2-pi-11-sqrt11. btw, what exactly do you mean by an algebraic Interpretation?2011-04-24
  • 0
    How do you get $\sqrt{11}$ on the RHS "by rewriting everything in terms of $\text{exp}(i\pi/11)$"?2011-05-24
  • 0
    @Américo: See [http://efreedom.com/Question/5-11246/Prove-Tan-Pi-11-Sin-Pi-11-Sqrt-11](http://efreedom.com/Question/5-11246/Prove-Tan-Pi-11-Sin-Pi-11-Sqrt-11)2011-05-27
  • 0
    Thanks for the information!2011-05-27

2 Answers 2