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)

$-\frac{1}{\sqrt{10}}$

Given $\tan x=\frac{3}{4}$ and $x$ lies in third quadrant $\cos x<0$

So ${\tan }^{2}x=\frac{9}{16}\Rightarrow {\sec }^{2}x-1=\frac{9}{16}$

$\Rightarrow {\sec }^{2}x=\frac{25}{16}$

$\Rightarrow {\cos }^{2}x=\frac{16}{25}$

$\Rightarrow \cos x=-\frac{4}{5}$

Now using $\cos x=2{\cos }^2\frac{x}{2}-1$

$\cos \frac{x}{2}=-\sqrt{\frac{1+\cos x}{2}}=-\sqrt{\frac{1-\frac{4}{5}}{2}}=-\frac{1}{\sqrt{10}}$

Note: If $x$ lies in the third quadrant, then $\frac{x}{2}$ will lie in $2nd$ quadrant