I Feel Nostalgic!

Algebra Level 3

Let \(\{a,b\}\) and \(\{m,n\}\) be consecutive terms (in that order) of 2 different geometric progressions with common ratio \(r_1>0\) and \(r_2>0\) respectively, such that \[(a-b)m=(a+b)n.\] Let \(z_1\) and \(z_2\) be \(2\) complex numbers such that \[\begin{align} \arg(z_1z_2)&=\arctan(r_1)+\arctan(r_2)=\theta\quad\quad (0\leq \theta\leq \pi), \\ \arg\left(\frac{z_1}{z_2}\right)&=\frac{\pi}9. \end{align}\]

Find \(2\arg(z_1)\).

Notation: The function \(\arg(z)\) denotes the argument of a complex number.

Clarification: Take the value of \(\pi=3.1415\).

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