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11 December, 10:09

How to find a power in rectangular form

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  1. 11 December, 10:11
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    You could just use the binomial theorem and expand it out then simplify it

    BUT ... the chances of messing up are pretty good ... so I think it's way easier to use De Moivre's Theorem to do it

    Drawing it on an argand diagram - √3 + i is in the 2nd quadrant

    mod (-√3 + i) = √ (3 + 1) = 2

    arg (-√3 + i) = π - arctan (1 / √3) = π - (π/6) = 5π/6

    so (-√3 + i) ^6 = {2 [cos (5π/6) + i sin (5π/6) ]}^6

    = 2^6 [cos (5π) + i sin (5π) ] ... [using De Moivre's theorem [r (cos θ + i sin θ]^n = r^n [cos (nθ) + i sin (nθ) ]

    = 64 [-1 + 0 i]

    = - 64
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