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# Find the principal values of the following: $tan^{-1} (-\sqrt3)$

$\begin{array}{1 1} \frac{\pi}{\sqrt 3} \\ \frac{-\pi}{\sqrt 3} \\ \frac{-\pi}{3} \\ \frac{\pi}{3} \end{array}$

Ans : $tan^{-1}tan \large\frac{-\pi}{3}=\large\frac{-\pi}{3}$

edited Mar 15, 2013

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• The range of the principal value of $\; tan^{-1}x$ is $\left [ \large-\frac{\pi}{2}, \large\frac{\pi}{2} \right ]$
Let $tan^{-1}-\sqrt 3 = x \Rightarrow tan (x) = -\sqrt 3$
We know that the range of the principal value of $\; tan^{-1}x$ is $\left [ \large-\frac{\pi}{2}, \large\frac{\pi}{2} \right ]$
Therefore, $tan (x) = -\sqrt 3 = - tan \large\frac{\pi}{3} z$$= tan \large\frac{-\pi}{3}$
$\Rightarrow x=\large\frac{-\pi}{3}$, where $x \;\epsilon\;$ $\left [ \large-\frac{\pi}{2}, \large\frac{\pi}{2} \right ]$
edited Apr 19 by meena.p