The polar form of 1−i=2(cos(−4π+isin(−4π)).
According to the De Moivre's Formula, all 3-th roots of a complex number
2(cos(−4π+isin(−4π)) are given by
(2)1/3(cos(3−π/4+2πk)+isin(3−π/4+2πk)),k=0,1,2 k=0:
(2)1/6(cos(3−π/4+2π(0))+isin(3−π/4+2π(0)))
=(2)1/6(cos(12π)−isin(12π))
k=1:
(2)1/6(cos(3−π/4+2π(1))+isin(3−π/4+2π(1)))
=(2)1/6(cos(127π)+isin(127π))
k=2:
(2)1/6(cos(3−π/4+2π(2))+isin(3−π/4+2π(2)))
=(2)1/6(cos(45π)+isin(45π))
=(2)−1/3(1+i)
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