Answer
$$\left( { - 2\sqrt 3 , - 2,3} \right)$$
Work Step by Step
$$\eqalign{
& {\text{We have the cylindrical coordinates }}\left( {4,\frac{{7\pi }}{6},3} \right) \cr
& \left( {r,\theta ,z} \right):{\text{ }}\left( {4,\frac{{7\pi }}{6},3} \right) \to r = 4,{\text{ }}\theta = \frac{{7\pi }}{6},{\text{ }}z = 3 \cr
& {\text{Cylindrical to rectangular}} \cr
& x = r\cos \theta \to x = 4\cos \left( {\frac{{7\pi }}{6}} \right) = - 2\sqrt 3 \cr
& y = r\sin \theta \to y = 4\sin \left( {\frac{{7\pi }}{6}} \right) = - 2 \cr
& z = z \to z = 3 \cr
& {\text{The rectangular coordinates are:}} \cr
& \left( { - 2\sqrt 3 , - 2,3} \right) \cr} $$
