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.

You can use Newton-Rapson with $\tan\left(x\right) - 1/3 = 0$  and the 'starting point' $x = \pi/6$:
\begin{align}
&\texttt{Clear[i, x];}
\\
&\texttt{x = Pi/6;}
\\ &
\\
&\texttt{For[i = 0, i < 5, ++i,}
\\
&\texttt{x -= (Tan[x] - 1/3)/Sec[x]^2; Print[N[x]]]}
\end{align}

\begin{align}
&\texttt{0.340586, 0.321873, 0.321751, 0.321751}\,,\ \color{red}{0.321751}
\end{align}
