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Question

cube roots of 𝑧 in polar form where 𝑧=2(cos(105∘)+𝑖sin(105∘)).

Write the complex number 5−3𝑖 in polar form, 𝑟(cos(𝜃)+𝑖sin(𝜃)), with 𝑟>0,0≤𝜃≤2𝜋.

Answer 1

a)

z = 2e^iθ

z^1/3 = 2^1/3e^iθ/3 = 2^1/3(cos^1/3(105°)+isin^1/3(105°)

b)

r = √(25+9) = √34

tan^-1(-3/5) = -30.96 --> θ=360-30.96 ≈ 329° = 1.8278π radians (Quadrant IV)

f(r,θ) = √34(cos(1.8278π)+isin(1.8278π)