List 10 If $a$, $b$, $c$ are all integers (where $0<c<90$), and make $\sqrt{9-8 \sin 50^{\circ}}=a+$ $b \sin c^{\circ}$, find the value of $\frac{a+b}{c}$.
Transform the expression, so that $9-8 \sin 50^{\circ}$ becomes a perfect square.
Solve $9-8 \sin 50^{\circ}=9+8 \sin 10^{\circ}-8 \sin 10^{\circ}-8 \sin 50^{\circ}=9+8 \sin 10^{\circ}-$ $8\left[\sin \left(30^{\circ}-20^{\circ}\right)+\sin \left(30^{\circ}+20\right)\right]=9+8 \sin 10^{\circ}-8 \cos 20^{\circ}=9+8 \sin 10^{\circ}-$ $8\left(1-2 \sin ^{2} 10^{\circ}\right)=16 \sin ^{2} 10^{\circ}+8 \sin 10^{\circ}+1=\left(4 \sin 10^{\circ}+1\right)^{2}$, so $a=1, b=4$, $c=10, \frac{a+b}{c}=\frac{1}{2}$.