# How do you evaluate arctan(2/5)?

Nov 27, 2016

$\arctan \left(\frac{2}{5}\right) \cong \frac{2}{5} - \frac{8}{375} + \frac{32}{15625} \cong 0.381$ rounded to the third decimal figure

#### Explanation:

The mac Laurin series for $\arctan \left(x\right)$ is

arctan(x) = x −x^3/3+x^5/5+ · · · +(−1)^n/(2n+1)x^(2n+1) + o(x^( 2n+2))

so $\arctan \left(\frac{2}{5}\right) = \frac{2}{5} - {\left(\frac{2}{5}\right)}^{3} / 3 + {\left(\frac{2}{5}\right)}^{5} / 5 + 0 \left({\left(\frac{2}{5}\right)}^{6}\right)$

$\arctan \left(\frac{2}{5}\right) \cong \frac{2}{5} - \frac{8}{375} + \frac{32}{15625} \cong 0.381$ rounded to the third decimal figure

Indeed $\tan \left(0.381\right) \cong 0.4 = \frac{2}{5}$