Can Arctan(11/2) Be Expressed in Terms of Pi?

[rbzima]

Can anyone express arctan(11/2) in terms of pi. Is there an easy way to do this?

 

[Gib Z]

I do not remember the proof, however I remember that arctan (1/2) is not expressible as a rational multiple of pi.

Now, looking at 11 squared plus 2 squared, we get 125, which is a nice cube. Hence let's find z where $$z^3 = 2 + 11i$$.

Expressing in Polar form, we get;
$$z^3 = \sqrt{125}\exp ( i \arctan (11/2)$$.

Since 125 is a nice cube, one value of z is $$\sqrt{5} \exp ( \frac{ i \arctan (11/2)}{3})$$

Some very hefty trig manipulations and converting back to Cartesian form gives 2+11i, cubing that we can easily verify the calculation. Essentially this implies that arctan (1/2) = 3 arctan (11/2).