De Moivre's Theorem

De Moivre's Theorem is one of the most important results in complex number theory. It provides a simple formula for raising a complex number to any integer power and has numerous applications, including finding trigonometric identities and computing roots of complex numbers.

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Statement of De Moivre's Theorem

For any integer $n$n (positive, negative, or zero) and any angle $\theta$θ:

$(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$(cosθ+isinθ)n=cos(nθ)+isin(nθ)

Equivalently, if $z = r(\cos\theta + i\sin\theta)$z=r(cosθ+isinθ), then:

$z^n = r^n(\cos(n\theta) + i\sin(n\theta))$zn=rn(cos(nθ)+isin(nθ))

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Proof by Induction (for positive integers)

Base case: $n = 1$n=1: $(\cos\theta + i\sin\theta)^1 = \cos\theta + i\sin\theta$(cosθ+isinθ)1=cosθ+isinθ ✓

Inductive step: Assume the result holds for $n = k$n=k:

$(\cos\theta + i\sin\theta)^k = \cos(k\theta) + i\sin(k\theta)$(cosθ+isinθ)k=cos(kθ)+isin(kθ)

Then for $n = k + 1$n=k+1:

$(\cos\theta + i\sin\theta)^{k+1} = (\cos(k\theta) + i\sin(k\theta))(\cos\theta + i\sin\theta)$(cosθ+isinθ)k+1=(cos(kθ)+isin(kθ))(cosθ+isinθ)

Using the multiplication rule (addition formulae):

$= \cos(k\theta + \theta) + i\sin(k\theta + \theta) = \cos((k+1)\theta) + i\sin((k+1)\theta)$=cos(kθ+θ)+isin(kθ+θ)=cos((k+1)θ)+isin((k+1)θ)

By induction, the result holds for all positive integers $n$n.

For negative integers, we use $z^{-n} = \frac{1}{z^n}$z−n=zn1​ and the result follows similarly.

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Computing Powers of Complex Numbers