Evaluate [ ^ -1 (3 5 ) ] - Vidyadip

Question

Evaluate $\sin \left[\cos ^{-1}\left(\frac{3}{5}\right)\right]$

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Answer

Let $x=\cos ^{-1}\left(\frac{3}{5}\right)\ldots(i)$
$ \therefore \cos x=\frac{3}{5}$
$\therefore \sin x=\sqrt{1-\cos ^2 x}$
$=\sqrt{1-\frac{9}{25}}$
$=\sqrt{\frac{16}{25}}$
$=\frac{4}{5}$
$\Rightarrow x=\sin ^{-1}\left(\frac{4}{5}\right) \ldots \ldots . .(ii) $
From (i) and (ii), we get
$ \sin \left[\cos ^{-1}\left(\frac{3}{5}\right)\right]=\sin \left[\sin ^{-1}\left(\frac{4}{5}\right)\right]$
$=\frac{4}{5} $

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