# If tanx =3/4, π < x <3 π /2, find the values of sin (x/2), cos (x/2) and tan (x/2)?

Sep 28, 2016

Given $\pi < x < \frac{3 \pi}{2} \mathmr{and} \tan x = \frac{3}{4}$

$\pi < x < \frac{3 \pi}{2}$

$\implies \frac{\pi}{2} < \frac{x}{2} < \frac{3 \pi}{4} \to \frac{x}{2} \in \text{ 2nd quadrant}$

This means

$\sin \left(\frac{x}{2}\right) \to + v e$

$\cos \left(\frac{x}{2}\right) \to - v e$

$\tan \left(\frac{x}{2}\right) \to - v e$

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

$\implies \frac{2 \tan \left(\frac{x}{2}\right)}{1 - {\tan}^{2} \left(\frac{x}{2}\right)} = \frac{3}{4}$

$\implies 8 \tan \left(\frac{x}{2}\right) = 3 - 3 {\tan}^{2} \left(\frac{x}{2}\right)$

$\implies 3 {\tan}^{2} \left(\frac{x}{2}\right) + 8 \tan \left(\frac{x}{2}\right) - 3 = 0$

$\implies 3 {\tan}^{2} \left(\frac{x}{2}\right) + 9 \tan \left(\frac{x}{2}\right) - \tan \left(\frac{x}{2}\right) - 3 = 0$

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

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

This means

$\tan \left(\frac{x}{2}\right) = \frac{1}{3} \to \text{not acceptable as } \tan \left(\frac{x}{2}\right) \to - v e$

So $\tan \left(\frac{x}{2}\right) \to - 3$

Now

cos(x/2)=1/sec(x/2)=-1/sqrt(1+tan^2(x/2)

$= - \frac{1}{\sqrt{1 + {\left(- 3\right)}^{2}}} = - \frac{1}{\sqrt{10}}$

Again

$\sin \left(\frac{x}{2}\right) = \tan \left(\frac{x}{2}\right) \times \cos \left(\frac{x}{2}\right)$

$= - 3 \times \left(- \frac{1}{\sqrt{10}}\right) = \frac{3}{\sqrt{10}}$