Inverse Trigonometric Functions Question 33
Question: $ \sin ( \frac{1}{2}{{\cos }^{-1}}\frac{4}{5} )= $
[Karnataka CET 2003]
Options:
A) $ \frac{1}{\sqrt{10}} $
B) $ -\frac{1}{\sqrt{10}} $
C) $ \frac{1}{10} $ .
D) $ -\frac{1}{10} $
Show Answer
Answer:
Correct Answer: A
Solution:
Let $ {{\cos }^{-1}}\frac{4}{5}=x $
Therefore $ \cos x=\frac{4}{5} $ …..(i) Now $ \sin ( \frac{1}{2}{{\cos }^{-1}}\frac{4}{5} )=\sin ( \frac{x}{2} ) $ …..(ii) From (i), $ \cos x=\frac{4}{5} $
Therefore $ 1-2{{\sin }^{2}}\frac{x}{2}=\frac{4}{5} $
Therefore $ 2{{\sin }^{2}}\frac{x}{2}=1-\frac{4}{5}=\frac{1}{5} $
$ \Rightarrow \sin \frac{x}{2}=\sqrt{\frac{1}{10}} $ .
BETA
