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Geometry Level 4

$\large\sum_{k=1}^{15} \textrm{cis}^{2k-1}\left( \frac{\pi}{36} \right)$

Compute the imaginary part of the summation above.

Notation

\frac{1}{4 \sin \frac{\pi}{36}}

\sin \frac{\pi}{36}

\frac{1}{4}

\frac{2+\sqrt{3}}{4 \sin \frac{\pi}{36}}

\frac{2-\sqrt{3}}{4 \sin \frac{\pi}{36}}