Fifteen, as shown in Figure 8, if two intersecting chords of a circle, with $\mathrm{B}$ on the smaller arc $\mathrm{AD}$, and the circle's radius is $5, \mathrm{BC} = 6$, $\mathrm{AD}$ is bisected by $\mathrm{BC}$. Also, from $\mathrm{A}$, the chord $\mathrm{AD}$ is the only one that can be bisected by $\mathrm{BC}$. This implies that the sine of the smaller arc $\mathrm{AB}$ is a rational number. If this rational number is expressed in its simplest form as $-\frac{m}{n}$, find $mn$.

将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
Solve as shown in Figure 9, let $\mathrm{AD}$ and $\mathrm{BC}$ intersect at $P$, connect $O P$, then $O P \perp A D$. Connect $\mathrm{OA}, \mathrm{O} 3, \mathrm{O} C$, draw $\mathrm{OE} \perp . \mathrm{BC}$ at $E$, and let $P A=P D=y, P E=$ $x$, then $B P=3-x$. Also, from the given conditions, $\mathrm{OE}=4$.
By the intersecting chords theorem
$$
\mathrm{AP} \cdot \mathrm{PD}=\mathrm{BP} \cdot \mathrm{PC} \text {, }
$$

which is
$$
y^{2}=(3-x)(3+x)=9-x^{2} .
$$

Since point $P$ is unique, the circle with $O A$ as the radius is tangent to $B C$, thus the angle between the tangent and the chord $\angle \mathrm{OPE}=\angle:$ :
Because $\cos \mathbf{A}=\frac{\rho \mathbf{A}}{0 \mathbf{A}}=\frac{Z}{5}$,
$$
\cos O P E=\frac{P E}{O P}=\frac{x}{\sqrt{16+x^{2}}},
$$

thus, $\frac{y}{5}=-\frac{x}{\sqrt{16+x^{2}}}$.
Squaring and substituting $y^{2}=9^{2}-x^{2}$, we can solve to get $x=2$.
Therefore, $C P=x+3=5$.
Thus, $\triangle \mathrm{CPO}$ is an isosceles triangle, $\angle S \mathrm{PO}=\angle C O P$.
Since $\angle \mathrm{A}+\angle \mathrm{AOP}=90^{\circ}$, so
$$
\angle \mathrm{COP}+\angle \mathrm{AOP}=90^{\circ} \text {, }
$$

thus $O A \perp O C$.
$$
\text { Therefore, } \begin{array}{l} 
\angle \mathrm{AOB}+\angle \mathrm{BO}=90^{\circ}, \\
{ }_{\sin} \mathrm{AOB}=\cos \mathrm{BOC} .
\end{array}
$$

And $\cos B O C=\frac{\mathrm{OB}^{2}+\mathrm{OC}^{2}-\mathrm{BC}^{3}}{2 \cdot \mathrm{OB} \cdot \mathbf{O C}}=\frac{7}{25}$.
which is a rational number,
$\therefore \sin A O B=\frac{7}{25}$ is also a rational number,

i.e., $\frac{\mathrm{m}}{\mathrm{n}}-=\frac{7}{25}, \mathrm{mn}=25 \times 7=175$.