Example 4 As shown in Figure 7, in the quadrilateral pyramid $P-ABCD$, $PA \perp$ base $ABCD$, $BC=CD=2$, $AC=4$, $\angle ACB=\angle ACD=\frac{\pi}{3}$, $F$ is the midpoint of $PC$, and $AF \perp PB$.
Find (1) the length of $PA$;
(2) the sine value of the dihedral angle $B-AF-D$.
(2016, Gansu Provincial Preliminary of the National High School Mathematics Competition)
【Analysis】(1) As shown in Figure 8, connect $B D$, intersecting $A C$ at point $O$.
Given $B C=C D=2$, we know that $\triangle B C D$ is an isosceles triangle.
Since $A C$ bisects $\angle B C D$, it follows that $A C \perp B D$.
Taking $O$ as the origin, the directions of $\overrightarrow{O B}$, $\overrightarrow{O C}$, and $\overrightarrow{A P}$ as the positive directions of the $x$-axis, $y$-axis, and $z$-axis respectively, we establish a spatial rectangular coordinate system $O-x y z$. Then
$$
O C=C D \cos \frac{\pi}{3}=1 .
$$

Given $A C=4$, we have $A O=A C-O C=3$.
Also, $O D=C D \sin \frac{\pi}{3}=\sqrt{3}$, thus
$$
\begin{array}{l}
A(0,-3,0), B(\sqrt{3}, 0,0), \\
C(0,1,0), D(-\sqrt{3}, 0,0) .
\end{array}
$$

Since $P A \perp$ the base $A B C D$, we can set $P(0,-3, z)$.
Given that $F$ is the midpoint of $P C$, we have $F\left(0,-1, \frac{z}{2}\right)$.
Also, $\overrightarrow{A F}=\left(0,2, \frac{z}{2}\right), \overrightarrow{P B}=(\sqrt{3}, 3,-z)$,
$A F \perp P B$,
thus $\overrightarrow{A F} \cdot \overrightarrow{P B}=0$
$\Rightarrow 6-\frac{z^{2}}{2}=0, z=2 \sqrt{3}$ (negative value discarded)
$\Rightarrow|\overrightarrow{P A}|=2 \sqrt{3}$
$\Rightarrow$ The length of $P A$ is $2 \sqrt{3}$.
(2) From (1), we know
$$
\begin{array}{l}
A(0,-3,0), B(\sqrt{3}, 0,0), C(0,1,0), \\
D(-\sqrt{3}, 0,0), F(0,-1, \sqrt{3}) .
\end{array}
$$

By property (7), the equation of plane $A B F$ is
$$
3 x-\sqrt{3} y+2 z-3 \sqrt{3}=0 \text {, }
$$

and the equation of plane $A F D$ is
$$
3 x+\sqrt{3} y-2 z+3 \sqrt{3}=0 .
$$

Thus, the cosine of the angle between the normal vectors of the two planes is
$$
\cos \theta=\frac{1}{8} \text {. }
$$

Therefore, the sine of the dihedral angle $B-A F-D$ is $\frac{\sqrt{63}}{8}$.