Question

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

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

Answer 1

The correct option is C

$\frac{4\sqrt{5}}{9}$

Explanation for the correct option:

The given trigonometric expression: $\sin \left[2{\cos }^{-1}\left(\frac{\sqrt{5}}{3}\right)\right]$.

It is known that, $\mathbf{2}{\mathbf{cos}}^{\mathbf{-}\mathbf{1}}\left(x\right)\mathbf{=}{\mathbf{sin}}^{\mathbf{-}\mathbf{1}}\left(2x\sqrt{1-x^2}\right)$.

Thus, $\sin \left[2{\cos }^{-1}\left(\frac{\sqrt{5}}{3}\right)\right]=\sin \left[{\sin }^{-1}\left\{2\left(\frac{\sqrt{5}}{3}\right)\sqrt{1-{\left(\frac{\sqrt{5}}{3}\right)}^2}\right\}\right]$

$$\begin{gathered} \Rightarrow \sin \left[2{\cos }^{-1}\left(\frac{\sqrt{5}}{3}\right)\right]=\frac{2\sqrt{5}}{3}\sqrt{1-\frac{5}{9}} \\ \Rightarrow \sin \left[2{\cos }^{-1}\left(\frac{\sqrt{5}}{3}\right)\right]=\frac{2\sqrt{5}}{3}\sqrt{\frac{9-5}{9}} \\ \Rightarrow \sin \left[2{\cos }^{-1}\left(\frac{\sqrt{5}}{3}\right)\right]=\frac{2\sqrt{5}}{3}\sqrt{\frac{4}{9}} \\ \Rightarrow \sin \left[2{\cos }^{-1}\left(\frac{\sqrt{5}}{3}\right)\right]=\frac{2\sqrt{5}}{3}\times \frac{2}{3} \\ \Rightarrow \sin \left[2{\cos }^{-1}\left(\frac{\sqrt{5}}{3}\right)\right]=\frac{4\sqrt{5}}{9} \end{gathered}$$

Therefore, the value of $\sin \left[2{\cos }^{-1}\left(\frac{\sqrt{5}}{3}\right)\right]$ is $\frac{4\sqrt{5}}{9}$.

Hence, the correct option is (C).