Answer
$sin~\frac{A}{2} = \sqrt{\frac{7}{8}}$
Work Step by Step
If $90^{\circ} \lt A \lt 180^{\circ}$, then the angle $A$ is in quadrant II. Then $45^{\circ} \lt \frac{A}{2} \lt 90^{\circ}$, so the angle $\frac{A}{2}$ is in quadrant I.
We can find the value of $sin~\frac{A}{2}$:
$sin~\frac{A}{2} = \sqrt{\frac{1-cos~A}{2}}$
$sin~\frac{A}{2} = \sqrt{\frac{1-(-\frac{3}{4})}{2}}$
$sin~\frac{A}{2} = \sqrt{\frac{(\frac{7}{4})}{2}}$
$sin~\frac{A}{2} = \sqrt{\frac{7}{8}}$
