# How do you evaluate the equation arctan( -sqrt3 / 3 )?

Mar 8, 2018

Converting the question in EQUATION FORM :
$\textcolor{red}{x} = \arctan \left(- \frac{\sqrt{3}}{3}\right) = - \frac{\pi}{6}$

#### Explanation:

color(red)((1)tan^(-1)(-X)=-tan^-1(X),X in R
color(red)((2)tan(pi/6)=1/sqrt3.
$\textcolor{red}{\left(3\right) {\tan}^{- 1} \left(\tan \left(X\right)\right) = X , \forall X \in \left(- \frac{\pi}{2} , \frac{\pi}{2}\right)}$
If the equation is, $\textcolor{red}{x} = {\tan}^{- 1} \left(- \frac{\sqrt{3}}{3}\right)$, then applying (1) we get $x = - {\tan}^{- 1} \left(\frac{\sqrt{3}}{3}\right) = - {\tan}^{- 1} \left(\frac{1}{\sqrt{3}}\right)$, now from (2) we get$x = - {\tan}^{- 1} \left(\tan \left(\frac{\pi}{6}\right)\right) , w h e r e , \frac{\pi}{6} \in \left(- \frac{\pi}{2} , \frac{\pi}{2}\right)$
$x = - \frac{\pi}{6} \to$ [ Applying (3) ]