De Moivre’s theorem

1 De Moivre’s theorem

We have seen, in Section 10.2 Key Point 7, that, in polar form, if $z = r (cos θ + i sin θ)$ and $w = t (cos ϕ + i sin ϕ)$ then the product $z w$ is:

$z w = r t (cos (θ + ϕ) + i sin (θ + ϕ))$

In particular, if $r = 1, t = 1$ and $θ = ϕ$ (i.e. $z = w = cos θ + i sin θ$ ), we obtain

${(cos θ + i sin θ)}^{2} = cos 2 θ + i sin 2 θ$

Multiplying each side of the above equation by $cos θ + i sin θ$ gives

${(cos θ + i sin θ)}^{3} = (cos 2 θ + i sin 2 θ) (cos θ + i sin θ) = cos 3 θ + i sin 3 θ$

on adding the arguments of the terms in the product.

Similarly

${(cos θ + i sin θ)}^{4} = cos 4 θ + i sin 4 θ .$

After completing $p$ such products we have:

${(cos θ + i sin θ)}^{p} = cos p θ + i sin p θ$

where $p$ is a positive integer.

In fact this result can be shown to be true for those cases in which $p$ is a negative integer and even when $p$ is a rational number e.g. $p = \frac{1}{2}$ .

Key Point 12

If $p$ is a rational number:

{(cos θ + i sin θ)}^{p} \equiv cos p θ + i sin p θ

This result is known as De Moivre’s theorem .

Recalling from Key Point 8 that $cos θ + i sin θ = e^{i θ}$ , De Moivre’s theorem is simply a statement of the laws of indices:

${(e^{i θ})}^{p} = e^{i p θ}$