Find the value of $\arctan(1/3)$ How can I calculate $\arctan\left({1\over 3}\right)$ in terms of $\pi$ ? I know that $\tan^2(\frac{\pi}{6})= {1\over3}$ but don't know if that helps in any way.

There is no simple way to do this.  That is because the answer is not some nice rational fraction times $\pi$.  Of course, you can look up (or use a calculator) to determine the $\arctan(1/3)$ (which equals $0.322$ radians or $18.435^\circ$ ) and then divide by $\pi$, but I don't think that is what you are looking for!
There is a way to represent the $\arctan$ using the series:
$$\arctan(x) = \frac{\pi}2 - \frac1x + \frac1{3x^3} - \frac1{5x^5} + \frac1{7x^7} - \cdots$$
However this is only usable for $x>1$ (and the similar series for $x < 1$ doesn't include $\pi$!)
The only other way to get $\pi$ in the "answer" would be to recognize that the $\arctan(x)$ can equal $\theta + n\cdot\pi$ (where $n = 0, \pm1,2,3,...$).
So the $\arctan(1/3) = 0.322 +n\cdot\pi  = (0.102 + n)\cdot\pi$
  (where $n = 0, \pm1,2,3,...$).
Not very satisfying.
