Answer
$\sqrt {\frac{3+2\sqrt 2}{6}}$
Work Step by Step
Given $y$ in Quadrant I and $csc(y)=3$, we have $sin(y)=\frac{1}{3}, cos(y)=\sqrt {1-sin^2y}=\frac{2\sqrt 2}{3}$
Use the Half-Angle Formula, $cos\frac{y}{2}=\sqrt {\frac{1+cos(y)}{2}}=\sqrt {\frac{1+\frac{2\sqrt 2}{3}}{2}}
=\sqrt {\frac{3+2\sqrt 2}{6}}$
