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

MCQ

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

A
$\frac{2 \pi}{3}$
B
$\frac{5 \pi}{6}$
✓
$\frac{3 \pi}{4}$
D
$\frac{3 \pi}{2}$

✓

Answer

Correct option: C.

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

c
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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