Write the expression as an algebraic expression in $u, u>0$. \[ \sin \left[\cos ^{-1}\left(\frac{3}{\sqrt{u^{2}+9}}\right)\right] \] \[ \sin \left[\cos ^{-1}\left(\frac{3}{\sqrt{u^{2}+9}}\right)\right]= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Rationalize all denominators.)

Step 1 ：Let's write the given expression as an algebraic expression in $u$, where $u>0$.

Step 2 ：The given expression is: $\sin \left[\cos ^{-1}\left(\frac{3}{\sqrt{u^{2}+9}}\right)\right]$

Step 3 ：This is a composition of trigonometric functions. We can simplify it using the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$.

Step 4 ：Let's let $\cos(x) = \frac{3}{\sqrt{u^2+9}}$.

Step 5 ：Then we can solve for $\sin(x)$ using the Pythagorean identity: $\sin(x) = \sqrt{1 - \cos^2(x)}$

Step 6 ：Substituting $\cos(x)$ into the equation, we get $\sin(x) = \sqrt{1 - \left(\frac{3}{\sqrt{u^2+9}}\right)^2}$

Step 7 ：Simplifying the above equation, we get $\sin(x) = \sqrt{\frac{u^2}{u^2 + 9}}$

Step 8 ：So, the simplified algebraic expression in $u$, where $u>0$, is $\boxed{\sqrt{\frac{u^{2}}{u^{2}+9}}}$