As shown in the figure, in planar quadrilateral $A B C D$, $A B = 8$, $C D = 3$, $A D = 5 \sqrt { 3 }$, $\angle A D C = 90 ^ { \circ }$, $\angle B A D = 30 ^ { \circ }$. Points $E$ and $F$ satisfy $\overrightarrow { A E } = \frac { 2 } { 5 } \overrightarrow { A D }$ and $\overrightarrow { A F } = \frac { 1 } { 2 } \overrightarrow { A B }$. Fold $\triangle A E F$ along $E F$ to $\triangle P E F$ such that $P C = 4 \sqrt { 3 }$. (1) Prove: $E F \perp P D$; (2) Find the sine of the dihedral angle between plane $P C D$ and plane $P B F$.
(1) From $AB = 8, AD = 5\sqrt{3}, AE = 2\sqrt{3}, AF = 4$, and $\angle BAD = 30°$, by the law of cosines $EF = 2$, so $AE^2 + EF^2 = AF^2$, thus $AE \perp EF$, i.e., $EF \perp AD$. Therefore $EF \perp PE$ and $EF \perp DE$, so $EF \perp$ plane $PDE$, hence $EF \perp PD$. (2) $\frac{8\sqrt{65}}{65}$
As shown in the figure, in planar quadrilateral $A B C D$, $A B = 8$, $C D = 3$, $A D = 5 \sqrt { 3 }$, $\angle A D C = 90 ^ { \circ }$, $\angle B A D = 30 ^ { \circ }$. Points $E$ and $F$ satisfy $\overrightarrow { A E } = \frac { 2 } { 5 } \overrightarrow { A D }$ and $\overrightarrow { A F } = \frac { 1 } { 2 } \overrightarrow { A B }$. Fold $\triangle A E F$ along $E F$ to $\triangle P E F$ such that $P C = 4 \sqrt { 3 }$.\\
(1) Prove: $E F \perp P D$;\\
(2) Find the sine of the dihedral angle between plane $P C D$ and plane $P B F$.