Answer
$tan(\frac{\theta}{2}) = 2$
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 $\frac{\theta}{2}$ is in quadrant I.
If the hypotenuse is 5, and the adjacent side has a length of 3, then the length of the opposite side is $\sqrt{5^2-3^2} = 4$
$tan(\frac{\theta}{2}) = \frac{sin~\theta}{1+cos~\theta}$
$tan(\frac{\theta}{2}) = \frac{\frac{4}{5}}{1+(-\frac{3}{5})}$
$tan(\frac{\theta}{2}) = \frac{(\frac{4}{5})}{(\frac{2}{5})}$
$tan(\frac{\theta}{2}) = 2$
