Question

The value of $\sin \left[2 Cos^{-1} \frac {\sqrt {5}}{3} \right]$is

• A. $\frac {2\sqrt {5}}{3} $
• B. $\frac {\sqrt {5}}{3} $
• C. $\frac {2\sqrt {5}}{9} $
• D. $\frac {4\sqrt {5}}{9} $

Answer 1

Answer: (D)

We have, $\sin\left[2 \cos^{-1} \frac{\sqrt{5}}{3}\right] $
$ =\sin\left[\cos^{-1} \left(2 \left(\frac{\sqrt{5}}{3}\right)^{2} -1\right)\right] $
$ \left[\because 2 \cos^{-1} x = \cos^{-1} \left(2x^{2} -1\right)\right] $
$=\sin\left[\cos^{-1} \left(\frac{1}{9}\right)\right] $
$=\sin\left[\sin^{-1} \sqrt{1- \left(\frac{1}{9}\right)^{2}}\right] $
$ \left[\because \cos^{-1} x =\sin^{-1} \left(\sqrt{1-x^{2}}\right)\right] $
$=\sin\left[\sin^{-1} \sqrt{\frac{80}{81}}\right] $
$= \frac{\sqrt{80}}{9} $
$ = \frac{4\sqrt{5}}{9} $