Question

If tan x= 3/4 ,π < x < 3π/2, find the value of sin x/2, cos x/2 and the tan x/2 .​

Answer 1

$\underline{\huge{Answer:-}}$

$\\ \\ \large\mathcal{Since \:\pi < \frac{3\pi}{2},cos \: x \: is \: negative. }$

$\large \mathcal{ Also \: \: \frac{\pi}{2} < \frac{x}{2} < \frac{3\pi}{4}. }$

$\large \mathtt{Therefore, \: sin \frac{x}{2 } \: is \: positive \: cos \frac{x}{2} \: is \: negative. }$

Nᴏᴡ,

$\implies \: \large \mathcal{ {sec}^{2}x = 1 + {tan}^{2} x = 1 + \frac{9}{16} = \frac{25}{16} }$

$\implies \: \large \mathcal{ {cos}^{2} x \: = \frac{16}{25} \: or \: cos \: x = - \frac{ 4}{5} } \: </u></em><em><u>$

$\implies \: \large \mathcal{2 \: {sin}^{2} \: \frac{x}{2} = 1 \: cos \: x \: 1 + \frac{4}{5} = \frac{9}{5} . }$

$\implies \: \large \mathcal{ {sin}^{2} \: \frac{x}{2} = \frac{9}{10} }$

$\implies \: \large \mathcal{or \: \: sin \frac{x}{2} = \frac{3}{ \sqrt{10} } </u></em><em><u> }$

$\implies \: \large \mathcal{2 {cos}^{2} \frac{x}{2} = 1 + cos \: x \: = 1 - \frac{4}{5} = \frac{1}{5} }$

$\implies \: \large \mathcal{ {cos}^{2} \frac{x}{2} = \frac{1}{10} }$

$\implies \: \large \mathcal{cos \frac{x}{2} = - \frac{1}{ \sqrt{10} } }$

$\Large \fbox \mathtt{Hence, \: \: \: tan \frac{x}{2} = \frac{sin \frac{x}{2} }{cos \frac{x}{2} } = \frac{3}{ \sqrt{10} } \times ( \frac{ - \sqrt{10} }{1} ) = - 3.}$