Question

Assertion (A) The value of $\cos ^{-1}\left[\cos \left(\frac{13 \pi}{6}\right)\right]$ is $\frac{\pi}{6}$.
Reason (R) $\cos ^{-1}(\cos x)=x ; x \in[0, \pi]$.

• A. Both A and R are correct; R is the correct explanation of A
• B. Both A and R are correct; R is not the correct explanation of A
• C. A is correct; R is incorrect
• D. R is correct; A is incorrect

Answer 1

Answer: (A)

Let $\theta=\cos ^{-1}\left[\cos \left(\frac{13 \pi}{6}\right)\right] \neq \frac{13 \pi}{6} \left(\because \frac{13 \pi}{6} \notin[0, \pi]\right)$
Now, $ \theta=\cos ^{-1}\left[\cos \left(2 \pi+\frac{\pi}{6}\right)\right]$
$\left(\because\right.$ range of $\cos ^{-1} x$ is $\left.[0, \pi]\right)$
$=\cos ^{-1}\left[\cos \frac{\pi}{6}\right] (\because \cos [2 \pi+x]=\cos x)$
$=\frac{\pi}{6} \left\{\because \cos ^{-1}(\cos x)=x\right.$, if $\left.x \in[0, \pi]\right\}$
$\therefore$ Both $A$ and $R$ are correct; $R$ is correct explanation of $A$.