Question

tan x = 3/4, π < x < 3π/2
Find the value of sin x/2, cos x/2, tan x/2​

Answer 1

Given,

$\sf \dashrightarrow tan \: x = \dfrac{3}{4}, x \in \bf {III}_{Q}$

Since,

$\sf \dashrightarrow {sec}^{2} x = 1 + {tan}^{2} x$

$\sf \dashrightarrow 1 + \bigg( \dfrac{3}{4} \bigg)^{2}$

$\sf \dashrightarrow 1 + \dfrac{9}{16}$

$\sf \dashrightarrow {sec}^{2} x = \dfrac{25}{16}$

$\sf \dashrightarrow sec \: x = - \dfrac{5}{4} \: \: \: (\therefore \bf {III}_{Q})$

$\sf \dashrightarrow cos \: x = - \dfrac{4}{5}$

$\:$

Since,

$\sf \dashrightarrow \pi < x < \dfrac{3 \pi}{2}$

$\sf \dashrightarrow \dfrac{\pi}{2} < \dfrac{x}{2} < \dfrac{3 \pi}{4}$

$\sf \dashrightarrow \dfrac{x}{2} \in {Q}_{2}$

$\sf \dashrightarrow cos \dfrac{x}{2} = - \sqrt{\dfrac{1 + cos \: x}{2}}$

$\sf \dashrightarrow - \sqrt{\dfrac{1 + \bigg(\dfrac{-4}{5} \bigg)}{2}}$

$\sf \dashrightarrow - \sqrt{\dfrac{\dfrac{1}{5}}{2}}$

$\sf \dashrightarrow - \dfrac{1}{\sqrt{10}}$

$\sf \dashrightarrow sin \dfrac{x}{2} = \sqrt{\dfrac{1 - cos \: x}{2}}$

$\:$

$\sf \dashrightarrow \sqrt{\dfrac{1 - \bigg( \dfrac{-4}{5} \bigg)}{2}}$

$\sf \dashrightarrow \sqrt{\dfrac{9}{10}} = \dfrac{3}{\sqrt{10}}$

$\sf \dashrightarrow tan \dfrac{x}{2} = \dfrac{sin \dfrac{x}{2}}{cos \dfrac{x}{2}}$

$\sf \dashrightarrow \dfrac{\dfrac{+3}{\sqrt{10}}}{\dfrac{-1}{\sqrt{10}}} = - 3$