Question 15.5.3: If cos x=-1/3 and x lies in Quadrant III, find the values of......

Since x lies in Quadrant III, we have

(ii) 2 \cos ^{2} \frac{x}{2}=(1+\cos x)=\left(1-\frac{1}{3}\right)=\frac{2}{3} \\  \\\begin{array}{l}\Rightarrow \cos ^{2} \frac{x}{2}=\frac{1}{3} \\  \\\Rightarrow \cos \frac{x}{2}=-\frac{1}{\sqrt{3}}=\frac{-1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}=-\frac{\sqrt{3}}{3} .\end{array}

(iii) \tan \frac{x}{2}=\frac{\sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)}=\left(\frac{\sqrt{6}}{3} \times \frac{3}{-\sqrt{3}}\right)=-\sqrt{2} .