# How do you calculate tan ^-1 (-1/2)?

Nov 27, 2016

${\tan}^{- 1} \left(- \frac{1}{2}\right) = - \frac{1}{2} + \frac{1}{24} - \frac{1}{160} = - 0.465$
rounded to the third decimal figure

#### Explanation:

the Mac-laurin expansion is

${\tan}^{- 1} \left(x\right) = x - {x}^{3} / 3 + {x}^{5} / 5 + o \left({x}^{6}\right)$

by replacing $x = - \frac{1}{2}$ and taking only the first three expansion's terms we get

${\tan}^{- 1} \left(- \frac{1}{2}\right) = - \frac{1}{2} + \frac{1}{3 \cdot {2}^{3}} - \frac{1}{5 \cdot {2}^{5}} = - 0.46456 \ldots .$ that rounded up to the third decimal figure becomes $- 0.465$

indeed $\tan \left(- 0.465\right) = - 0.502$ rounded to the third decimal figure