Find the principal value of
Question:

Find the principal value of $\cos ^{-1}\left(-\frac{1}{\sqrt{2}}\right)$

Solution:

Let $\cos ^{-1}\left(-\frac{1}{\sqrt{2}}\right)=y$. Then, $\cos y=-\frac{1}{\sqrt{2}}=-\cos \left(\frac{\pi}{4}\right)=\cos \left(\pi-\frac{\pi}{4}\right)=\cos \left(\frac{3 \pi}{4}\right)$.

We know that the range of the principal value branch of $\cos ^{-1}$ is $[0, \pi]$ and

$\cos \left(\frac{3 \pi}{4}\right)=-\frac{1}{\sqrt{2}}$

Therefore, the principal value of $\cos ^{-1}\left(-\frac{1}{\sqrt{2}}\right)$ is $\frac{3 \pi}{4}$.

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