Question
In: Computer Science
Complex number
What are the values of all the cube roots (Z = -4^3 - 4i)?
Solutions
Expert Solution
z = 8 cis (2πk-5π/6) for k = 1,2,3.
z1/3 = 2 cis (2πk/3 - 5π/18) for k = 1,2,3.
= 2 cos(π/18) + 2i sin(π/18)
or 2 cos(7π/18) + 2i sin(7π/18)
or 2 cos(13π/18) + 2i sin(13π/18)
r =(((-4)(3)^(1/2))^2 + ((-4)^2))^(1/2) = 8
θ = arctan (-4/(-4(3)^(1/2) = 7∏/6.
roots = 2 (cos (7∏/18 + 12n∏/18) = i sin(7∏/18 + 12n∏/18), n = 0, 1, 2
r =(((-4)(3)^(1/2))^2 + ((-4)^2))^(1/2) = 8
θ = arctan (-4/(-4(3)^(1/2) = 7∏/6.
roots = 2 (cos (7∏/18 + 12n∏/18) = i sin(7∏/18 + 12n∏/18), n = 0, 1, 2
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