Find the principal value of ^ -1 (1 2 ) - Vidyadip

Question

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

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Answer

Suppose that $\cos ^{-1}\left(\frac{1}{\sqrt{2}}\right)=y, \Rightarrow \cos y=\frac{1}{\sqrt{2}}=\cos \frac{\pi}{4}$ We know that the range of principal value branch of $\cos ^{-1} x$ is $[0, \pi]$ and $\cos y=\cos \frac{\pi}{4}$. So, the principal value of $\cos ^{-1}\left(\frac{1}{\sqrt{2}}\right)$ will be $\frac{\pi}{4}$.

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