Simplifying a difficult trigonometric sum

Question

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

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"?

cvgmt · Accepted Answer · 2021-07-29 08:34:56Z

11

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

Sqrt[11]

answered Jul 29, 2021 at 8:34

cvgmt

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$\begingroup$ Oh gee. How nice. And quick. Thanks. ($\checkmark$). And what about the negative value? $\endgroup$
– David G. Stork
Commented Jul 29, 2021 at 8:37
$\begingroup$ Tan[(3 π)/11] + 4 Sin[(2 π)/11] // N equal to 3.31662. It is a positive number. $\endgroup$
– cvgmt
Commented Jul 29, 2021 at 8:41
$\begingroup$ But the video found $\pm \sqrt{11}$. Anyway, good enough. Again, thanks. $\endgroup$
– David G. Stork
Commented Jul 29, 2021 at 8:42
3

$\begingroup$ @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. $\endgroup$
– Michael Seifert
Commented Jul 29, 2021 at 14:12
8

$\begingroup$ Also Tan[(3*Pi)/11] + 4*Sin[(2*Pi)/11] // RootReduce $\endgroup$
– Bob Hanlon
Commented Jul 29, 2021 at 14:39

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