Choose the correct answer from the given four options. The value … - Vidyadip

Question

Choose the correct answer from the given four options.
The value of the expression $2\sec^{-1}2+\sin^{-1}\Big(\frac{1}{2}\Big)$ is:

$\frac{\pi}{6}$
$\frac{5\pi}{6}$
$\frac{7\pi}{6}$
$1$

✓

Answer

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

Solution:
We have, $2\sec^{-1}2+\sin^{-1}\Big(\frac{1}{2}\Big)=2\sec^{-1}\sec\frac{\pi}{3}+\sin^{-1}\sin\frac{\pi}{6}$
$=2\frac{\pi}{3}+\frac{\pi}{6}$
$[\because\ \sec^{-1}(\sec\text{x})=\text{x and }\sin^{-1}(\sin\text{x})=\text{x}]$
$=\frac{4\pi+\pi}{6}=\frac{5\pi}{6}$

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