Step-by-Step Solution

1. Evaluate $\sin^{-1}(\frac{1}{\sqrt{2}})$:

   Since $\sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$, we have $\sin^{-1}(\frac{1}{\sqrt{2}}) = \frac{\pi}{4}$.

2. Evaluate $\cos^{-1}(\frac{\sqrt{3}}{2})$:

   Since $\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$, we have $\cos^{-1}(\frac{\sqrt{3}}{2}) = \frac{\pi}{6}$.

3. Evaluate $\cos^{-1}(0)$:

   Since $\cos(\frac{\pi}{2}) = 0$, we have $\cos^{-1}(0) = \frac{\pi}{2}$.

4. Substitute these values into the expression:

   $3\sin^{-1}(\frac{1}{\sqrt{2}})+2\cos^{-1}(\frac{\sqrt{3}}{2})+\cos^{-1}(0) = 3(\frac{\pi}{4}) + 2(\frac{\pi}{6}) + \frac{\pi}{2}$

5. Simplify the expression:

   $= \frac{3\pi}{4} + \frac{\pi}{3} + \frac{\pi}{2} = \frac{9\pi + 4\pi + 6\pi}{12} = \frac{19\pi}{12}$