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Geometry Level 4

\[\large \prod _{ n=1 }^{ 15 }{ \sin { \left( \frac { n\pi }{ 31 } \right) } } =\frac { \sqrt { \alpha } }{ { 2 }^{ \beta } }\]

If the above equation holds true for coprime positive integers \(\alpha\) and \(\beta\), find \(\alpha +\beta\).


This problem was created in celebration of the Rio 2016 Summer Olympics (Games of the XXXI Olympiad).
Image Credit: Olympic.org.
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