Find the principal value of cos ^{-1}left(-fr | vedclass

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Question

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

Vedclass · Accepted Answer

$\frac{3 \pi}{4}$

Vedclass · Answer

(C) Let $y = \cos ^{-1}\left(-\frac{1}{\sqrt{2}}ight)$.Then $\cos y = -\frac{1}{\sqrt{2}}$.Since $\cos \left(\frac{\pi}{4}ight) = \frac{1}{\sqrt{2}}$,we have $\cos y = -\cos \left(\frac{\pi}{4}ight)$.Using the identity $\cos (\pi - 	heta) = -\cos 	heta$,we get $\cos y = \cos \left(\pi - \frac{\pi}{4}ight) = \cos \left(\frac{3 \pi}{4}ight)$.The range of the principal value branch of $\cos ^{-1}$ is $[0, \pi]$.Since $\frac{3 \pi}{4} \in [0, \pi]$,the principal value of $\cos ^{-1}\left(-\frac{1}{\sqrt{2}}ight)$ is $\frac{3 \pi}{4}$.

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

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