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)

$\frac{1}{\sqrt{10}}$

Let ${\cos }^{-1}\frac{4}{5}$$=\theta$

$\cos \theta =\frac{4}{5}$

So, $1-2{\sin }^2\left(\frac{\theta }{2}\right)=\frac{4}{5}$

$\Rightarrow 2{\sin }^2\left(\frac{\theta }{2}\right)=\frac{1}{5}$

${\sin }^2\left(\frac{\theta }{2}\right)=\frac{1}{10}$

$\sin \left(\frac{\theta }{2}\right)=\frac{1}{\sqrt{10}}$