# Simplifying a difficult trigonometric sum

As this video shows,

$$\tan \left( \frac{3 \pi}{11} \right) + 4 \sin \left( \frac{2 \pi}{11} \right) = \sqrt{11} .$$

The direct and obvious methods for Mathematica to find this solution do not work:

Simplify[Tan[(3 \[Pi])/11]+ 4 Sin[(2 \[Pi])/11]]


or

TrigReduce[Tan[(3 \[Pi])/11] + 4 Sin[(2 \[Pi])/11]]


and the obvious application of FullSimplify, TrigExpand, and so on.

Is there any way to get Mathematica to find this reduction without having to impose clever "human knowledge"?

Tan[(3 π)/11] + 4 Sin[(2 π)/11] // ToRadicals // FullSimplify

Sqrt[11]
• Oh gee. How nice. And quick. Thanks. ($\checkmark$). And what about the negative value? Jul 29 at 8:37
• Tan[(3 π)/11] + 4 Sin[(2 π)/11] // N equal to 3.31662. It is a positive number. Jul 29 at 8:41
• But the video found $\pm \sqrt{11}$. Anyway, good enough. Again, thanks. Jul 29 at 8:42
• @DavidG.Stork: If you watch the last bit of the video (starting around 25:00), the presenter points out that since $3\pi/11$ is in the first quadrant, the quantity on the left-hand side must be positive. So the video's final answer is $+\sqrt{11}$ only. Jul 29 at 14:12
• Also Tan[(3*Pi)/11] + 4*Sin[(2*Pi)/11] // RootReduce Jul 29 at 14:39