Question

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

Answer 1

$\tan x=\frac{3}{4}$

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

$\Rightarrow \tan 2\left(\frac{x}{2}\right)=\frac{3}{4}$

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

$\Rightarrow \tan x/2=t$

$\Rightarrow \frac{2t}{1-t^2}=\frac{3}{4}$

$\Rightarrow 8t=3-3t^2$

$\Rightarrow 3t^2+8t-3=0$

$\Rightarrow 3t^2-9t-t-3=0$

$\Rightarrow 3t(t+3)-1(t+3)=0$

$\Rightarrow t=\frac{1}{3}t=-3$

$\Rightarrow \tan x/2=\frac{1}{3}$ or $\tan x/2=-3$

since $\frac{x}{2}\in (\frac{\pi }{2},\frac{3\pi }{4})$

$\Rightarrow \frac{\pi }{2}=2^{nd}$ quadrant

$\Rightarrow \tan x/2=$ negative

$\therefore \tan x/2=-3$

$\therefore =sinx/2=\frac{3}{\sqrt{10}}$

$\cos x/2=\frac{-1}{\sqrt{10}}$

$\tan x/2=-3$