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Model Paper 5 — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsModel Paper 52 Marks
Question
Using the principal values, write the value of $\cos ^{-1}\left(\frac{1}{2}\right)+2 \sin ^{-1}\left(\frac{1}{2}\right)$.
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Answer

We have, $\cos ^{-1}\left(\frac{1}{2}\right)=\cos ^{-1}\left(\cos \frac{\pi}{3}\right)$
$=\frac{\pi}{3}\left[\because \frac{\pi}{3} \in[0, \pi]\right]$
Also $\sin ^{-1}\left(-\frac{1}{2}\right)=\sin ^{-1}\left(-\sin \frac{\pi}{6}\right)$
$=\sin ^{-1}\left(\sin \left(-\frac{\pi}{6}\right)\right)$
$=-\frac{\pi}{6}\left[\because-\frac{\pi}{6} \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\right]$
$\therefore \cos ^{-1}\left(\frac{1}{2}\right)-2 \sin ^{-1}\left(-\frac{1}{2}\right)=\frac{\pi}{3}-2\left(-\frac{\pi}{6}\right)$
$=\frac{\pi}{3}+\frac{\pi}{3}=\frac{2 \pi}{3}$]

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