The process of mathematical induction can be used to prove a very important theorem in mathematics known as De Moivre's theorem. If the complex number z = r(cos α + i sin α), then
The preceding pattern can be extended, using mathematical induction, to De Moivre's theorem.
If z = r(cos α + i sin α), and n is a natural number, then
Example 1: Write 
in the form s + bi.
First determine the radius:
Since cos α = 
and sin α = ½, α must be in the first quadrant and α = 30°. Therefore,
Example 2: Write 
in the form a + bi.
First determine the radius:
Since cos 
and sin 
, α must be in the fourth quadrant and α = 315°. Therefore,
Problems involving powers of complex numbers can be solved using binomial expansion, but applying De Moivre's theorem is usually more direct.
De Moivre's theorem can be extended to roots of complex numbers yielding the nth root theorem. Given a complex number z = r(cos α + i sinα), all of the nth roots of z are given by
where k = 0, 1, 2, …, (n − 1)
If k = 0, this formula reduces to
This root is known as the principal nth root of z. If α = 0° and r = 1, then z = 1 and the nth roots of unity are given by
where k = 0, 1, 2, …, ( n − 1)
Example 3: What are each of the five fifth‐roots of 
expressed in trigonometric form?
Since cos 
and sin α = ½, α is in the first quadrant and α = 30°. Therefore, since the sine and cosine are periodic,
and applying the nth root theorem, the five fifth‐roots of z are given by
where k = 0, 1, 2, 3, and 4
Thus the five fifth‐roots are
Observe the even spacing of the five roots around the circle in Figure 1 .
Figure 1
Drawing for Example 3.