Uses of De Moivre’s Theorem

De Moivre’s Theorem is used for various purposes. Some of its most important uses are,

• Finding the Roots of Complex Numbers.
• Finding the relationships between Powers of Trigonometric Functions and Trigonometric Angles.
• Solving the Power of Complex Numbers.

Now, let’s learn about them with the help of examples.

Finding the Roots of Complex Numbers

The polar form of the complex number is,

z = r(cos x + i sin x)

Then for n^th root of the complex number

z^1/n = r^1/n(cos x + i sin x)^1/n

⇒ z^1/n = r^1/n[cos (x + 2kπ)/n + i sin (x + 2kπ)/n]

Where k = 0, 1, 2, 3, …

Power of Complex Numbers

We can easily solve the power of Complex numbers using De Moivre’s Theorem. This can be understood using the example as follows,

Example: Evaluate (√3 + i)²⁰⁰

Solution:

Let, z = √3 + i comparing with z = x + iy

x = √3, y = 1

Also, z = r(cos θ + i sin θ)

r = √(x² + y²) = √[(√3)² + 1²]

r = 2

θ = tan^-1(y/x) = tan^-1(1/√3) = π/6

z = r(cos θ + i sin θ)

⇒ z = 2(cos π/6 +i.sin π/6)

⇒ z²⁰⁰ = [2(cos π/6 +i.sin π/6)]²⁰⁰

⇒ z²⁰⁰ = [2]²⁰⁰[(cos π/6 +i.sin π/6)]²⁰⁰

Using DeMoivre’s Theorem

z²⁰⁰ = [2]²⁰⁰[(cos 200π/6 +i.sin 200π/6)]

⇒ z²⁰⁰ = [2]²⁰⁰[-1/2 – i√3/2]

⇒ z²⁰⁰ = [2]²⁰⁰[1/2 + i√3/2]

Read More,

• Complex Number Power Formula
• Polar and Exponential Forms of Complex Numbers
• Z Bar in Complex Numbers
  
• Is Every Real Number a Complex Number? 

De Moivre’s theorem is one of the fundamental theorem of complex numbers which is used to solve various problems of complex numbers. This theorem is also widely used for solving trigonometric functions of multiple angles. DeMoivre’s Theorem is also called “De Moivre’s Identity” and “De Moivre’s Formula”. This theorem gets its name from the name of its founder the famous mathematician De Moivre.

In this article, we will learn about De Moivre’s Theorem, its proof, some examples based on the theorem, and others in detail.