Write the value

Question:

Write the value of $\cos ^{-1}\left(\cos \frac{5 \pi}{4}\right)$.

Solution:

$\cos ^{-1}\left(\cos \frac{5 \pi}{4}\right) \neq \frac{5 \pi}{4}$ as $\frac{5 \pi}{4}$ does not lie between 0 and $\pi$.

We have

$\cos ^{-1}\left(\cos \frac{5 \pi}{4}\right)=\cos ^{-1}\left\{\cos \left(2 \pi-\frac{3 \pi}{4}\right)\right\}=\cos ^{-1}\left\{\cos \left(\frac{3 \pi}{4}\right)\right\}=\frac{3 \pi}{4}$

 

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