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