So many sides

Algebra Level 4

If line segments are drawn from the origin of the complex plane to each of the solutions of \(x^n = 1\), and consecutive solutions are also connected by line segments we obtain \(n\) isosceles triangles in a regular \(n\)-gon. Let \(a+bi\) be the solution with the smallest possible positive argument. If \(\arctan{\left(\dfrac{b}{a}\right)} = \dfrac{\pi}{90}\), then find the value of \(n\).

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