Answer
$\dfrac{\sqrt {10}}{4}$
Work Step by Step
We are given that $f(\dfrac{\alpha}{2})=\sin (\dfrac{\alpha}{2})$
The half-angle Identity for sine can be expressed as:
$\sin {(\dfrac{ \alpha}{2})}=\pm \sqrt{\dfrac{1 -\cos ( \alpha)}{2}} $
Because the angle $\dfrac{ \alpha}{2}$ lies in the first quadrant, we take the positive sign for sine.
We have: $\cos \alpha=-\dfrac{1}{4}$. Now, plug these values into the following identity:
$\sin {(\dfrac{ \alpha}{2})}=\sqrt{\dfrac{1 -(\dfrac{-1}{4})}{2}} =\sqrt {\dfrac{4+1}{8}}=\dfrac{\sqrt 5}{2 \sqrt 2}=\dfrac{\sqrt {10}}{4}$
