Find the principal values: ^ -1 ( -3 ) - Vidyadip

Question

Find the principal values:
$\tan^{-1}\left({-\sqrt{3}}\right)$

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Answer

$\text{Let Y}=\tan^{-1}\left(-\sqrt{3}\right)$, $\text{where} -\frac{{\pi}}{2}<\text{Y}<\frac{{\pi}}{2}$
$\therefore \ \tan\text{Y}=-\sqrt{3}$, $\text{where} -\frac{{\pi}}{2}<\text{Y}<\frac{{\pi}}{2}$
$\therefore \ \tan\text{Y}=-\tan \frac{{\pi}}{3}=\tan \left(-\frac{{\pi}}{3}\right) \text{where}-\frac{{\pi}}{2}<\text{Y}<\frac{{\pi}}{2}$
$\therefore \ \text{Y}=-\frac{{\pi}}{3}$
$\therefore$ required principal value $=-\frac{{\pi}}{3}.$

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