Find the Exact Value cos(arcsin(( square root of 2)/3))

Draw a triangle in the plane with vertices ${\displaystyle (\sqrt{1^2-{\left(\frac{\sqrt{2}}{3}\right)}^2},\frac{\sqrt{2}}{3})}$, ${\displaystyle (\sqrt{1^2-{\left(\frac{\sqrt{2}}{3}\right)}^2},0)}$, and the origin. Then ${\displaystyle \arcsin \left(\frac{\sqrt{2}}{3}\right)}$ is the angle between the positive x-axis and the ray beginning at the origin and passing through ${\displaystyle (\sqrt{1^2-{\left(\frac{\sqrt{2}}{3}\right)}^2},\frac{\sqrt{2}}{3})}$. Therefore, ${\displaystyle \cos \left(\arcsin \left(\frac{\sqrt{2}}{3}\right)\right)}$ is ${\displaystyle \sqrt{\frac{7}{9}}}$.

Rewrite ${\displaystyle \sqrt{\frac{7}{9}}}$ as ${\displaystyle \frac{\sqrt{7}}{\sqrt{9}}}$.