Appeasing the Cherry Blossom Horde

By the sine rule, the ratio is $\sin\alpha / \sin\beta$ where $\alpha < \beta$ are uniform on $[0, \pi/2]$. Conditioning on $\alpha < \beta$ and integrating gives $\tfrac{8}{\pi^2} \ln 2 \approx 0.5618$, matched by a million-sample Monte Carlo simulation at $0.5619$. Full derivation and code in the PDF.