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Find the principal value of each of the following :

Question:

Find the principal value of each of the following :

$\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)$

 

Solution:

$\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)=\cos ^{-1}\left(\cos \left(2 \pi-\frac{5 \pi}{6}\right)\right)$

[Formula: $\cos (2 \pi-x)=\cos (x)$, as $\cos$ has a positive vaule in the fourth quadrant. ]

$=\cos ^{-1}\left(\cos \frac{5 \pi}{6}\right)\left[\right.$ Formula: $\cos ^{-1}(\cos x)=x$

$=\frac{5 \pi}{6}$

 

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