Question

If tan $x = \frac{3}{4},$ $ \pi < x < \frac {3\pi}{2}$ then the value of $cos \frac {x}{2}$ is

• A. $\frac {3}{\sqrt {10}}$
• B. $-\frac {3}{\sqrt {10}}$
• C. $-\frac {1}{\sqrt {10}}$
• D. $\frac {1}{\sqrt {10}}$

Answer 1

Answer: (C)

Given, $\tan x=\frac{3}{4}, \pi< x< \frac{3 \pi}{2}$
$\Rightarrow \cos x=-\frac{4}{5}$
$\because 1+\cos x=2 \cos ^{2}\left(\frac{x}{2}\right)$
$\therefore 1-\frac{4}{5}=2 \cos ^{2}\left(\frac{x}{2}\right)$
$\Rightarrow \frac{1}{5}=2 \cos ^{2}\left(\frac{x}{2}\right)$
$\Rightarrow \cos ^{2}\left(\frac{x}{2}\right)=\frac{1}{10}$
$\Rightarrow \cos \left(\frac{x}{2}\right)=\pm \frac{1}{\sqrt{10}}$
$\cos \left(\frac{x}{2}\right)=-\frac{1}{\sqrt{10}}$
$\left[\because \frac{\pi}{2}<\frac{x}{2}<\frac{3 \pi}{4}\right]$