\(sin\left(\frac{1}{2}cos^{-1}\frac{4}{5}\right)=?\) A. \(\frac{1}

1.	\(sin\left(\frac{1}{2}cos^{-1}\frac{4}{5}\right)=?\) A. \(\frac{1}{\sqrt{5}}\) B. \(\frac{2}{\sqrt{5}}\) C. \(\frac{1}{\sqrt{10}}\) D. \(\frac{2}{\sqrt{10}}\)
Answer» Correct Answer is (C) \(\sqrt{\frac{1}{10}}\) Let x =\(cos^{-1}\frac{4}{5}\) ⇒ cos x =\(\frac{4}{5}\) Therefore \(sin\left(\frac{1}{2}cos^{-1}\frac{4}{5}\right)\) becomes sin(\(\frac{1}{2}\)x),i.e sin (\(\frac{x}{2}\)) We know that sin (\(\frac{x}{2}\)) = \(\sqrt{\frac{1-cosx}{2}}\) =\(\sqrt{\frac{1-\frac{4}{5}}{2}}\) =\(\sqrt{\frac{\frac{2}{5}}{2}}\) sin(\(\frac{x}{2}\))=\(\sqrt{\frac{1}{10}}\)

\(sin\left(\frac{1}{2}cos^{-1}\frac{4}{5}\right)=?\) A. \(\frac{1}{\sqrt{5}}\) B. \(\frac{2}{\sqrt{5}}\) C. \(\frac{1}{\sqrt{10}}\) D. \(\frac{2}{\sqrt{10}}\)

Answer»

Correct Answer is (C) \(\sqrt{\frac{1}{10}}\)

Let x =\(cos^{-1}\frac{4}{5}\)

⇒ cos x =\(\frac{4}{5}\)

Therefore \(sin\left(\frac{1}{2}cos^{-1}\frac{4}{5}\right)\) becomes sin(\(\frac{1}{2}\)x),i.e sin (\(\frac{x}{2}\))

We know that sin (\(\frac{x}{2}\)) = \(\sqrt{\frac{1-cosx}{2}}\)

=\(\sqrt{\frac{1-\frac{4}{5}}{2}}\)

=\(\sqrt{\frac{\frac{2}{5}}{2}}\)

sin(\(\frac{x}{2}\))=\(\sqrt{\frac{1}{10}}\)