ArcTan expressed as a radian fraction?

Question

This type of answer is what I'm looking for:

In[58]:= ArcTan @ 1
Out[58]=  π/4 

This is what mathematica gives me:

In[59]:= ArcTan@2
Out[59]= ArcTan[2]

Is it possible to express ArcTan in terms of $\pi$? I understand some fractions would be hairy.

I am using Mathematica 8.

Answer 1

Maybe this

HoldForm[Pi]  (1/Pi ArcTan@2.)

or if you want a nicer way

Rationalize /@ (HoldForm[Pi] N@(1/Pi ArcTan@Range[5]))

Edit

The latter method works well in cases when there is a rational fraction of $\pi$ :

Rationalize /@ (HoldForm[Pi] N @ (1/Pi ArcTan @ { Sqrt[1 - 2/Sqrt[5]], 2 - Sqrt[3], 
                                                  1/Sqrt[3], Sqrt[3], 1}))

To sum up : Mathematica does what it should do, namely ArcTan[2] is not a rational fraction of $\pi$ and that's why it returns ArcTan[2] unlike in case ArcTan[1]. The above method is to express ArcTan[x] in terms of a real multiple of $\pi$.

If you want to get back what you have evaluated you shoud use ReleaseHold, e.g.

Tan @ ReleaseHold @ %

Answer 2

Short answer: no, ArcTan[2] is not fraction of $\pi$. But this is more of a mathematics question than pertaining to Mathematica.

If you want to “check” that the result is not expressable as a fraction of $\pi$, you can check for the continued fraction reprentation of $\arctan(2)/\pi$, and see that it does not seem to converge:

Table[FromContinuedFraction@ContinuedFraction[ArcTan[2]/\[Pi], n], {n, 20}]