Find the principal values of the following:    9. $\cos^{-1}\left(-\frac{1}{\sqrt2} \right )$

Let us assume $\cos^{-1}\left(-\frac{1}{\sqrt2} \right ) = x$;

Then we have $\cos x = \left ( \frac{-1}{\sqrt 2} \right )$

or

$-\cos (\frac{\pi}{4}) = \left ( \frac{-1}{\sqrt 2} \right )$  ,

$\cos (\pi - \frac{\pi}{4}) = \cos (\frac{3\pi}{4})$.

And we know the range of principal values of $\cos^{-1}$ is $[0,\pi]$.

So, the only principal value which satisfies $\cos^{-1}\left(-\frac{1}{\sqrt2} \right ) = x$ is $\frac{3\pi}{4}$.

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