Question

The value of $\sin \left(\frac{1}{2}{\cos }^{-1}\frac{4}{5}\right)$ is?

• A. $+\frac{1}{\sqrt{1}0}$
• B. $-\frac{1}{\sqrt{1}0}$
• C. Both (A) and (B)
• D. None of these

Answer 1

Answer: (C)

Both (A) and (B)

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