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

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

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  • Created on: 06-05-15 17:33
Preview of FP2 - Complex Numbers - Proof of de Moivre's Theorem

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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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