Question

$sin\left(\frac{1}{2}co{s}^{-1}\frac{4}{5}\right)=$

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

Answer 1

The correct option is A $\frac{1}{\sqrt{10}}$

Let $0\le x\le \pi$
So, $co{s}^{-1}\frac{4}{5}=x\Rightarrow cosx=\frac{4}{5}$ ....(i)
Now $sin\left(\frac{1}{2}co{s}^{-1}\frac{4}{5}\right)=sin\left(\frac{x}{2}\right)$ ....(ii)
From (i), $cosx=\frac{4}{5}\Rightarrow 1-2si{n}^2\frac{x}{2}=\frac{4}{5}$
$\Rightarrow 2si{n}^2\frac{x}{2}=1-\frac{4}{5}=\frac{1}{5}\Rightarrow sin\frac{x}{2}=\sqrt{\frac{1}{10}}$.