SOLUTION
We want to evaluate
[tex]\begin{gathered} \cos (\frac{x}{2}) \\ \text{If }\cos (x)=-\frac{2}{5}\text{ and in the third quadrant } \end{gathered}[/tex]
Using the half angle formula, we have
[tex]\begin{gathered} \cos (\frac{x}{2})=\pm\sqrt[]{\frac{1+\cos(x)}{2}} \\ =\pm\sqrt[]{\frac{1-\frac{2}{5}}{2}} \\ =\pm\sqrt[]{\frac{\frac{3}{5}}{2}} \\ =\pm\sqrt[]{\frac{3}{10}} \end{gathered}[/tex]
Now since the angle (x) is the third quadrant, that means
[tex]\begin{gathered} \frac{x}{2}\text{ should fall under the second quadrant } \\ \text{and in the second quadrant, } \\ cos\theta\text{ is negative} \end{gathered}[/tex]
Hence the answer becomes
[tex]\cos (\frac{x}{2})=-\sqrt[]{\frac{3}{10}}[/tex]
