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
First determine the radius:
in the form s + bi.
Since cos α =
Therefore,
and sin α = ½, α must be in the first quadrant and α = 30°.
Example 2: Write
First determine the radius:
in the form a + bi.
Since cos
315°. Therefore,
and sin
, α must be in the fourth quadrant and α =
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
form?
expressed in trigonometric
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