# FP2 - Complex Numbers - Proof of de Moivre's Theorem

Proof of de Moivre’s theorem when n is a positive integer

- Created by: chloeeyay
- Created on: 06-05-15 17:33

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Proof of de Moivre's theorem when n is a positive integer

Using proof by induction

W hen n = 1 LHS = [r(cos + isin)]1 = r(cos + isin)

W hen n = 1 RHS = r1(cos1 + isin1) = r(cos + isin)

As LHS = RHS, de Moivre's theorem is true for n = 1

Assume that de Moivre's theorem is true for n = k, k +

[r(cos + isin)]k+1 = [ r(cos + isin)]k× r(cos + isin)

k

[ rk(cosk + isink)] × r(cos + isin)

rk+1(cosk + isink)(cos + isin)

rk+1(cos(k + ) + isin(k + ))

rk+1(cos (k + 1) ) + (isin(k + 1))

Therefore, de Moivre's theorem is true when n = k + 1

If it's true for n = k + 1 then it must be true for n = k

As de Moivre's theorem is true for n = 1, it is also now true for all n1 by mathematical

induction.

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