Question

$If\tan x=\frac{3}{4},\pi <x<\frac{3\pi }{2},$ find the values of $sin\frac{x}{2},\cos \frac{x}{2},\tan \frac{x}{2}$

Answer 1

$\tan x=\frac{2\tan \frac{x}{2}}{1-{\tan }^2\frac{x}{2}}=\frac{3}{4}$

$\Rightarrow 3{\tan }^2\frac{x}{2}+8\tan \frac{x}{2}-3=0$

$\Rightarrow (3\tan \frac{x}{2}-1)(\tan \frac{x}{2}+3)=0$

$\Rightarrow \tan \frac{x}{2}=\frac{1}{3}$, as $\pi <x<\frac{3\pi }{2}$

$\Rightarrow \sin \frac{x}{2}=\frac{1}{\sqrt{1^2+3^2}}=\frac{1}{\sqrt{10}}$

$\Rightarrow \cos \frac{x}{2}=\frac{3}{\sqrt{10}}$