Know your continued fractions

Geometry Level 4

\[8\sin { \frac { \pi }{ 10 } \cos { \frac { \pi }{ 5 } } } +4\sin { \frac { \pi }{ 10 } } +1=\alpha +\cfrac { 1 }{ \alpha +\cfrac { 1 }{ \alpha +\cfrac { 1 }{ \alpha +\ddots } } }\] Find \(\alpha\).


This problem is original.

Image credit: Wikimedia Levochik

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