Find the principal values of the following:

    9. \cos^{-1}\left(-\frac{1}{\sqrt2} \right )

Answers (1)

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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