Posted by **Mathslover** on Wednesday, May 29, 2013 at 11:54pm.

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.

- Complex angles -
**Steve**, Thursday, May 30, 2013 at 10:54am
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

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