Practice: Finding the sum of several cosines

Geometry Level 3

What value of \(N\) in the interval \( [0, 90] \) satisfies the following equation: \[ \begin{align} \frac { \sin N ^ \circ } { 2 \sin 5 ^ \circ } - \frac {1}{2} = & \cos 10 ^ \circ + \cos 20 ^ \circ + \cos 30^ \circ + \cos 40 ^ \circ \\ &+ \cos 50 ^ \circ + \cos 60^ \circ + \cos 70 ^ \circ + \cos 80 ^ \circ\, ?\\ \end{align} \]