Q73 / 180 · KCET 2007 · Mathematics
The value of $\sin \left[2 \cos ^{-1} \frac{\sqrt{5}}{3}\right]$ is
- A$\frac{\sqrt{5}}{3}$
- B$\frac{2 \sqrt{5}}{3}$
- Correct answerC$\frac{4 \sqrt{5}}{9}$
- D$\frac{2 \sqrt{5}}{9}$
Solution
We have, $\sin \left[2 \cos ^{-1} \frac{\sqrt{5}}{3}\right]$
$=\sin \left[\cos ^{-1}\left(2 \cdot\left(\frac{\sqrt{5}}{3}\right)^{2}-1\right)\right]$
$\left[\because 2 \cos ^{-1} x=\cos ^{-1}\left(2 x^{2}-1\right)\right]$
$=\sin \left[\cos ^{-1}\left(\frac{1}{9}\right)\right]$
$=\sin \left[\sin ^{-1} \sqrt{1-\left(\frac{1}{9}\right)^{2}}\right]$
$\left[\because \cos ^{-1} x=\sin ^{-1}\left(\sqrt{1-x^{2}}\right)\right]$
$=\sin \left[\sin ^{-1} \sqrt{\frac{80}{81}}\right]$
$$
=\frac{\sqrt{80}}{9}=\frac{4 \sqrt{5}}{9}
$$
Hence, the correct answer is option: 'C'.