Summing sines

Geometry Level 4

Given that \( \displaystyle \sum_{k=1}^{35} \sin(5\cdot k^\circ)= \tan \left( \dfrac mn \right) \), where \(m\) and \(n\) are coprime positive integers satisfying \( \dfrac mn < 90 \), find \(m+n\).

Clarification: The angles are measured in degrees.

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