# Complex Angles

Algebra Level 3

There are four complex fourth roots to the number $$4-4\sqrt{3}i$$. These can be expressed in polar form as

$z_1 = r_1\left(\cos \theta_1 +i\sin \theta_1 \right)$ $z_2 =r_2\left(\cos\theta_2+i\sin\theta_2\right)$ $z_3 = r_3\left(\cos\theta_3+i\sin\theta_3 \right)$ $z_4 = r_4\left(\cos\theta_4+i\sin\theta_4\right),$

where $$r_i$$ is a real number and $$0^\circ \leq \theta_i < 360^\circ$$. What is the value of $$\theta_1 + \theta_2 + \theta_3 + \theta_4$$ (in degrees)?

Details and assumptions

$$i$$ is the imaginary unit satisfying $$i^2=-1$$.

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