De Moivre's Theorem Calculator

Formula

The formula to calculate De Moivre's Theorem is:

$z^n = r^n \left( \cos(n\theta) + i \sin(n\theta) \right)$

Where:

$z$ is a complex number
$n$ is a positive integer exponent
$r$ is the magnitude (absolute value) of the complex number $z$
$\theta$ is the argument (angle) of the complex number $z$

What is De Moivre's Theorem?

De Moivre's Theorem is a formula in the field of complex numbers that connects trigonometry and complex numbers. Named after Abraham de Moivre, a French mathematician, this theorem is particularly useful in simplifying and solving equations involving complex numbers. The theorem states that for any real number $x$ and any integer $n$ , the formula $(\cos x + i \sin x)^n = \cos nx + i \sin nx$ holds true, where $i$ is the imaginary unit. This theorem allows for the easy calculation of powers and roots of complex numbers and is also used in deriving trigonometric identities. It is a key tool in the study of trigonometry, calculus, and complex number theory.

Example Calculation

Let's assume the following values:

Magnitude (r) = 2
Argument (θ) = π/4 radians
Exponent (n) = 3

Using the formula to calculate De Moivre's Theorem:

$z^n = 2^3 \left( \cos(3 \cdot \frac{\pi}{4}) + i \sin(3 \cdot \frac{\pi}{4}) \right)$

Calculating the components:

$r^n = 2^3 = 8$

$\cos(3 \cdot \frac{\pi}{4}) = \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2}$

$\sin(3 \cdot \frac{\pi}{4}) = \sin(\frac{3\pi}{4}) = \frac{\sqrt{2}}{2}$

So the result is:

$z^n = 8 \left( -\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right) = -4\sqrt{2} + 4\sqrt{2}i$

The result of De Moivre's Theorem is $-4\sqrt{2} + 4\sqrt{2}i$ .