Question:
Mark the tick against the correct answer in the following:
$\sin \left(\frac{1}{2} \cos ^{-1} \frac{4}{5}\right)=?$
A. $\frac{1}{\sqrt{5}}$
B. $\frac{2}{\sqrt{5}}$
C. $\frac{1}{\sqrt{10}}$
D. $\frac{2}{\sqrt{10}}$
Solution:
To Find: The value of $\sin \left(\frac{1}{2} \cos ^{-1} \frac{4}{5}\right)$
Let $x=\cos ^{-1} \frac{4}{5}$
$\Rightarrow \cos x=\frac{4}{5}$
Therefore $\sin \left(\frac{1}{2} \cos ^{-1} \frac{4}{5}\right)$ becomes $\sin \left(\frac{1}{2} x\right), i . e \sin \left(\frac{x}{2}\right)$
We know that $\sin \left(\frac{x}{2}\right)=\sqrt{\frac{1-\cos x}{2}}$
$=\sqrt{\frac{1-\frac{4}{5}}{2}}$
$=\sqrt{\frac{\frac{2}{5}}{2}}$
$\sin \left(\frac{x}{2}\right)=\frac{1}{\sqrt{10}}$