<s>[INST] 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.
 [/INST] For the evaluation of $\tan^{-1}(x)$, you can also use Padé approximants such as
$$\frac{ x+\frac{4 x^3}{15}} { 1+\frac{3 x^2}{5}}\tag 1$$
$$\frac{x+\frac{11 x^3}{21} }  {1+\frac{6 x^2}{7}+\frac{3 x^4}{35} } \tag 2$$
$$\frac{x+\frac{7 x^3}{9}+\frac{64 x^5}{945}}  {1+\frac{10 x^2}{9}+\frac{5 x^4}{21} } \tag 3$$
$$\frac{x+\frac{34 x^3}{33}+\frac{x^5}{5} }  { 1+\frac{15 x^2}{11}+\frac{5 x^4}{11}+\frac{5 x^6}{231}} \tag 4$$
Using the above formulae, you would get $\frac{139}{432}$, $\frac{250}{777}$, $\frac{20806}{64665}$, $\frac{19593}{60895}$ which are $\approx 0.3217592593$, $\approx 0.3217503218$, $\approx 0.3217505606$,  $\approx 0.3217505542$ while the "exact" value would be $\approx 0.3217505544$.
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