Question

$\sin \left(\frac{1}{2}{\cos }^{-1}\frac{4}{5}\right)$ is equal to:

• A. $-\frac{1}{\sqrt{10}}$
• B. $\frac{1}{\sqrt{10}}$
• C. $-\frac{1}{10}$
• D. $\frac{1}{10}$

Answer 1

Answer: (B)

Given: $\sin \left(\frac{1}{2}{\cos }^{-1}\frac{4}{5}\right)$
Now, Let $\frac{4}{5}=\cos 2\theta$
$\therefore \sin \left(\frac{1}{2}{\cos }^{-1}\frac{4}{5}\right)=\sin \left(\frac{1}{2}{\cos }^{-1}\cos 2\theta \right)$
$=\sin \left(\frac{1}{2}\times 2\theta \right)=\sin \theta =\sqrt{\frac{1-\cos 2\theta }{2}}$
$=\sqrt{\frac{1-\frac{4}{5}}{2}}=\frac{1}{\sqrt{10}}$