Answer
$\frac{\sqrt 7}{3}$
Work Step by Step
Step 1. Letting $sin^{-1}\frac{\sqrt 2}{3}=t$, we have $sin(t)=\frac{\sqrt 2}{3}$
Step 2. Let $r=3, y=\sqrt 2$, and $x=\sqrt {r^2-y^2}=\sqrt 7$.
Step 3. Sides $x,y,r$ form a right triangle with angle $t$ facing $y$.
Step 4. We have $cos(t)=\frac{x}{r}=\frac{\sqrt 7}{3}$ or $cos(sin^{-1}\frac{\sqrt 2}{3})=\frac{\sqrt 7}{3}$