De Moivre's Theorem

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.