Question 15.5.2: If tan x=3/4 and π<x<3π/2, find the values of (i) sin......
If \tan x=\frac{3}{4} and \pi<x<\frac{3 \pi}{2} , find the values of
(i) \sin \frac{x}{2}, \quad (ii) \cos \frac{x}{2} ,\quad (iii) \tan \frac{x}{2} .
Since x lies in Quadrant III, we have \cos x<0 .
Now, \tan x=\frac{3}{4} \Rightarrow \sec ^{2} x=\left(1+\tan ^{2} x\right)=\left(1+\frac{9}{16}\right)=\frac{25}{16}\\ \\\begin{array}{l}\Rightarrow \cos ^{2} x=\frac{1}{\sec ^{2} x}=\frac{16}{25} \\ \\\Rightarrow \cos x=-\sqrt{\frac{16}{25}}=\frac{-4}{5} .\end{array}
Also, \pi<x<\frac{3 \pi}{2} \Rightarrow \frac{\pi}{2}<\frac{x}{2}<\frac{3 \pi}{4} \\ \\\begin{array}{l}\Rightarrow \frac{x}{2} \text { lies in Quadrant II } \\ \\\Rightarrow \sin \frac{x}{2}>0 \text { and } \cos \frac{x}{2}<0 .\end{array}
(i) 2 \sin ^{2} \frac{x}{2}=(1-\cos x)=\left(1+\frac{4}{5}\right)=\frac{9}{5} \\ \\\begin{array}{l}\Rightarrow \sin ^{2} \frac{x}{2}=\frac{9}{10}\\ \\\Rightarrow \sin \frac{x}{2}=+\sqrt{\frac{9}{10}}=\frac{3}{\sqrt{10}} \quad\left[\because \sin \frac{x}{2}>0\right] .\end{array}
(ii) 2 \cos ^{2} \frac{x}{2}=(1+\cos x)=\left(1-\frac{4}{5}\right)=\frac{1}{5} \\ \\\begin{array}{l}\Rightarrow \cos ^{2} \frac{x}{2}=\frac{1}{10}\\ \\\Rightarrow \cos \frac{x}{2}=-\sqrt{\frac{1}{10}}=\frac{-1}{\sqrt{10}} \quad\left[\because \cos \frac{x}{2}<0\right] .\end{array}
(iii) \tan \frac{x}{2}=\frac{\sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)}=\left(\frac{3}{\sqrt{10}} \times \frac{\sqrt{10}}{-1}\right)=-3 .