Answer
$cos~\frac{\theta}{2} = \frac{1}{2}$
Work Step by Step
If $90^{\circ} \lt \theta \lt 180^{\circ}$, then the angle $\theta$ is in quadrant II. Then $45^{\circ} \lt \frac{\theta}{2} \lt 90^{\circ}$, so the angle $\frac{\theta}{2}$ is in quadrant I.
We can find the value of $cos~\frac{\theta}{2}$:
$cos~\frac{\theta}{2} = \sqrt{\frac{1+cos~\theta}{2}}$
$cos~\frac{\theta}{2} = \sqrt{\frac{1+(-\frac{1}{2})}{2}}$
$cos~\frac{\theta}{2} = \sqrt{\frac{(\frac{1}{2})}{2}}$
$cos~\frac{\theta}{2} = \sqrt{\frac{1}{4}}$
$cos~\frac{\theta}{2} = \frac{1}{2}$
