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 = 1 2 .

Key Point 12

If p is a rational number:

( cos θ + i sin θ ) p 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