Answer
$ \sqrt{\dfrac{(5 +\sqrt 5)}{10}}$
Work Step by Step
We have: $\sin \theta=\dfrac{2}{\sqrt 5}$ and $\cos \theta=-\dfrac{1}{\sqrt 5}$
The half-angle Identity for sine can be expressed as:
$\sin {(\dfrac{\theta}{2})}=\pm\sqrt{\dfrac{1-\cos{\theta}}{2}}$
Because the angle $\dfrac{\theta}{2}$ lies in the first quadrant, we take the positive sign for cosine.
$\sin {(\dfrac{\theta}{2})}=\sqrt{\dfrac{1 - (\dfrac{-1}{\sqrt 5})}{2}} = \sqrt{\dfrac{\sqrt 5 + 1}{2 \sqrt 5 }} = \sqrt{\dfrac{(5 +\sqrt 5)}{10}}$