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{ α 3 β 5 = 1 α 7 β 2 = 1 \begin{cases} \alpha^3 \beta^5 = 1 \\ \alpha^7 \beta^2 = 1 \end{cases} {α3β5=1α7β2=1
Let α = cos θ 1 + i sin θ 1 \alpha = \cos \theta_1 + i \sin \theta_1 α=cosθ1+isinθ1and β = cos θ 2 + i sin θ 2 \beta = \cos \theta_2 + i \sin \theta_2 β=cosθ2+isinθ2be thecomplex numberssatisfying the system above, where 0 < θ 1 0 < \theta_1 0<θ1and θ 2 < π 2 < \ \ theta_2 frac{\pi}{2} θ2<2π.
If θ 1 θ 2 = a b \frac{\theta_1}{\theta_2} = \frac{a}{b} θ2θ1=ba, where a a aand b b bare coprimepositive integers, compute a + b . a+b. a+b.
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