11. (14 points) As shown in Figure 1, let $S-ABCD$ be a pyramid with a height of 3, where the base $ABCD$ is a square with side length 2, and the projection of vertex $S$ on the base is the center of the square $ABCD$. $K$ is the midpoint of edge $SC$, and a plane passing through $AK$ intersects segments $SB$ and $SD$ at points $M$ and $N$ respectively. ($M$ and $N$ can be the endpoints of the segments).
(1) Find the sine value of the angle formed by line $AK$ and plane $SBC$; when $M$ is the midpoint of $SB$, find the volume of the pyramid $S-AMKN$.
11. (1) Let the angle between $A K$ and plane $S B C$ be $\theta$.

Since $S C=\sqrt{3^{2}+(\sqrt{2})^{2}}=\sqrt{11}$, therefore, $C K=\frac{\sqrt{11}}{2}$,
and $\cos \angle S C A=\frac{\sqrt{2}}{\sqrt{11}}=\frac{\sqrt{22}}{11}$.
Then $A K^{2}=A C^{2}+C K^{2}-2 A C \cdot C K \cos \angle S C A=\frac{27}{4}$.
Thus, $A K=\frac{3 \sqrt{3}}{2}$.
Also, $V_{=\text {:Rens } S-A B C}=\frac{1}{3} \times \frac{1}{2} \times 2 \times 2 \times 3=2$
$=V_{\text {=.投的A } 1 \text { - SCB }}$,
then $h_{A}=\frac{6}{S_{\triangle S B C}}=\frac{3 \sqrt{10}}{5}$.
Therefore, $\sin \theta=\frac{h_{1}}{A K}=\frac{2 \sqrt{30}}{15}$.
(2) Let the projections of $N$ and $D$ on plane $S B C$ be $H$ and $H_{1}$, respectively.
Then $\frac{N H}{D H_{1}}=\frac{S N}{S D}$.
Also, $\frac{S_{\triangle S M K}}{S_{\triangle S B C}}=\frac{S M \cdot S K}{S B \cdot S C}$, hence
$=\frac{S M \cdot S K \cdot S N}{S B \cdot S C \cdot S D}$.
When $M$ is the midpoint of $S B$, $M K \parallel B C$.
Since $B C \parallel$ plane $S A D$, therefore,
$M K \parallel$ plane $S A D$,

i.e., $M K \parallel A N, M K \parallel A D$.
Thus, point $N$ coincides with $D$.
It is easy to find that the distance from point $M$ to plane $S A K$ is $\frac{\sqrt{2}}{2}$.
Also, $S_{\triangle S A K}=\frac{1}{2} S_{\triangle S A C}=\frac{3 \sqrt{2}}{2}$, hence
$V_{\text {四佼的S-AMKN }}=\frac{1}{3} \times \sqrt{2} S_{\triangle S M K}+\frac{1}{3} \times \frac{\sqrt{2}}{2} S_{\triangle S K K}$ $=\frac{\sqrt{2}}{2} S_{\triangle S U K}=\frac{\sqrt{3}}{2}$.