Gujarat BoardEnglish MediumSTD 12 ScienceMathsInverse Trigonometric Functions1 Mark
Question
Find the principal value of $\cos ^{-1}\left(-\frac{1}{\sqrt{2}}\right)$
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Answer
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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