$\left|BO\right|=5$

$\left|AO\right|=5\sqrt{3}$

Z tw. Pitagorasa w $\Delta SOB$:

$H^2+5^2=13^2$

$H^2+25=169|-25$

$H^2=144|\sqrt{},H>0$

$H=12$

Z tw. Pitagorasa w $\Delta SOA$:

$12^2+{\left(5\sqrt{3}\right)}^2={\left|AS\right|}^2$

${\left|AS\right|}^2=144+75=219|\sqrt{},\left|AS\right|>0$

$\left|AS\right|=\sqrt{219}$

$\cos \alpha =\frac{\left|AO\right|}{\left|AS\right|}=\frac{5\sqrt{3}}{\sqrt{219}}=\frac{5\cancel{\sqrt{3}}}{\cancel{\sqrt{3}}\cdot \sqrt{73}}=$

$=\frac{5}{\sqrt{73}}\cdot \frac{\sqrt{73}}{\sqrt{73}}=\underline{\underline{\frac{5\sqrt{73}}{73}}}$