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} \times2 \theta\right) = \sin\theta = \sqrt{\frac{1- \cos2\theta}{2}} $
$= \sqrt{\frac{1- \frac{4}{5}}{2}} = \frac{1}{\sqrt{10}} $