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The tangent plane at a point $P_0(f(u_0,v_0),g(u_0,v_0),h(u_0,v_0))$ on a parametrized surface $r(u,v)=f(u,v)ig(u,v)jh(u,v)k$ is the plane through $P_0$ normal to the vector $r_u(u_0,v_0)r_v(u_0,v_0),$ the cross product of the tangent vectors $r_u(u_0,v_0)$ and $r_v(u_0,v_0)$ at $P_0.$ Find an equation for the plane tangent to the surface at $P_0.$ Then find a Cartesian equation for the surface and sketch the surface and tangent plane together.

The circular cylinder $r(,z)=(3sin(2))i(6si{n}^2)jzk$ at the point $P_0(\frac{3\sqrt[\phantom{2}]{3}}{2},\frac{3}{2},2)$ corresponding to $(,z)=(\frac{}{6},2)$

An equation for the plane tangent to the surface at $P_0$ is $$

$($Type an equation using $x,y,$ and $z$ as the variables.$)$