Find the principal values: ^ -1 (-1 2 ) - Vidyadip

Question

Find the principal values:
$\cos^{-1}\bigg(-\frac{1}{\sqrt{2}}\bigg)$

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Answer

$\text{Let Y}=\cos^{-1}\bigg(-\frac{1}{\sqrt2}\bigg) \text{where}\ 0\leq\text{Y}\leq {\pi}$,
$ \therefore \cos\text{Y}=-\frac{1}{\sqrt{2}}\ \ \text{where}\ 0\leq\text{Y}\leq {\pi}$
$\therefore\cos\text{Y}=-\cos\frac{{\pi}}{4}=\cos\bigg({\pi}-\frac{{\pi}}{4}\bigg)=\cos\frac{3\pi}{4}$
$\therefore\text{Y}=\frac{3\pi}{4}$
$\therefore$ required principal value $=\frac{3\pi}{4} $

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