Complex angles
posted by Mathslover .
There are four complex fourth roots to the number 4−43√i. These can be expressed in polar form as
z1=r1(cosθ1+isinθ1)
z2=r2(cosθ2+isinθ2)
z3=r3(cosθ3+isinθ3)
z4=r4(cosθ4+isinθ4),
where ri is a real number and 0∘≤θi<360∘. What is the value of θ1+θ2+θ3+θ4 (in degrees)?
Details and assumptions
i is the imaginary unit satisfying i2=−1.

If the smallest angle is θ1, then the other angles are
θ1+90,θ1+180,θ1+270, so
θ1+θ2+θ3+θ4 = 4θ1 + 540
So, what is θ1?
Well, √i = (1+i)/√2, so
4−43√i = 4  (43+43i)/√2 = 40.27 cis 229.027
So, the 4th root with smallest angle is
2.52 cis 57.26
So, θ1+θ2+θ3+θ4 = 4(57.26) + 540 = 769 
Z_(1 ) and Z_(2 )are given by Z_1=5*j(〖60〗^0) Z_2=4*j〖45〗^0