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0
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The following polar grid is centered at the origin, with concentric circles of positive integer radii starting at $1,$ and consecutive rays through the origin with angles of $\pi/12$ between them:

[asy] size(200); pair A = (0,0); for (int i = 1; i < 6; ++i) { draw(Circle((0,0),i), linewidth(0.4)); } for(int i=0;i<360;i+=15) { draw(rotate(i)*((-5.0,0)--(5.0,0)), linewidth(0.4)); } pair A; A = 3*dir(45)^2*unit((1,1)); dot(A, red+linewidth(3.5)); label("$A$", A,N); [/asy]

For the point $A$ above, say that four possible pairs of polar coordinates for $A$ are

$(3, \theta_1), (3, \theta_2), (-3, \theta_3), (-3, \theta_4),$

where $\theta_1,\theta_2, \theta_3, \theta_4$ are distinct angles in radians between $0$ and $4\pi$. Enter

$\theta_1+ \theta_2, \theta_3 + \theta_4$

in that order.

Jul 30, 2021

$\theta + \theta_2 = 3 \pi$, $\theta_3 + \theta_4 = \frac{16 \pi}{3}$.