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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