Question

The value of $\sin \left(\frac{1}{4}{\sin }^{-1}\frac{\sqrt{63}}{8}\right)$ is

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{1}{2\sqrt{2}}$
• D. $\frac{1}{5}$

Answer 1

The correct option is C $\frac{1}{2\sqrt{2}}$

$\sin \left(\frac{1}{4}{\sin }^{-1}\frac{\sqrt{63}}{8}\right)$

Let $x={\sin }^{-1}\frac{\sqrt{63}}{8}$

$⟹\sin x=\frac{\sqrt{63}}{8}$

$⟹\cos x=\frac{\sqrt{64-63}}{8}=\frac{1}{8}$

$\cos \frac{x}{2}=\sqrt{\frac{1+\cos x}{2}}$

$=\sqrt{\frac{1}{2}\left(\frac{9}{8}\right)}$

$\cos \frac{x}{2}=\frac{3}{4}$

$⟹\cos \frac{x}{4}=\sqrt{\frac{1+\cos \frac{x}{2}}{2}}$

$=\sqrt{\frac{1}{2}(1+\frac{3}{4})}$

$=\sqrt{\frac{7}{8}}$

$⟹\sin \frac{x}{4}$

$=\sqrt{1-{\cos }^2\frac{x}{4}}$

$=\sqrt{1-\frac{7}{8}}$

$=\frac{1}{2\sqrt{2}}$