Complex Angles

Algebra Level 3

There are four complex fourth roots to the number 443i4-4\sqrt{3}i. These can be expressed in polar form as

z1=r1(cosθ1+isinθ1)z_1 = r_1\left(\cos \theta_1 +i\sin \theta_1 \right) z2=r2(cosθ2+isinθ2)z_2 =r_2\left(\cos\theta_2+i\sin\theta_2\right) z3=r3(cosθ3+isinθ3)z_3 = r_3\left(\cos\theta_3+i\sin\theta_3 \right) z4=r4(cosθ4+isinθ4),z_4 = r_4\left(\cos\theta_4+i\sin\theta_4\right),

where rir_i is a real number and 0θi<3600^\circ \leq \theta_i < 360^\circ. What is the value of θ1+θ2+θ3+θ4\theta_1 + \theta_2 + \theta_3 + \theta_4 (in degrees)?

Details and assumptions

ii is the imaginary unit satisfying i2=1i^2=-1.


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