Prime Power Roots

Algebra Level 3

{α3β5=1α7β2=1\begin{cases} \alpha^3 \beta^5 = 1 \\ \alpha^7 \beta^2 = 1 \end{cases}

Let α=cosθ1+isinθ1\alpha = \cos \theta_1 + i \sin \theta_1 and β=cosθ2+isinθ2\beta = \cos \theta_2 + i \sin \theta_2 be the complex numbers satisfying the system above, where 0<θ10 < \theta_1 and θ2<π2\theta_2 < \frac{\pi}{2}.

If θ1θ2=ab\frac{\theta_1}{\theta_2} = \frac{a}{b}, where aa and bb are coprime positive integers, compute a+b.a+b.

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