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March 27, 2017

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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 - ,

    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

  • Complex angles - ,

    Z_(1 ) and Z_(2 )are given by Z_1=5*j(-〖60〗^0) Z_2=4*j〖45〗^0

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