Enter your complex number in polar form and the exponent into the calculator to determine the result using De Moivre’s Theorem.

## De Moivre’s Theorem Calculation Formula

The following formula is used to calculate the powers of complex numbers using De Moivre’s Theorem.

(r(cos θ + i sin θ))^n = r^n (cos(nθ) + i sin(nθ))

Variables:

- r is the modulus of the complex number
- θ is the argument (angle) of the complex number
- n is the exponent
- cos and sin are trigonometric functions

To calculate the power of a complex number using De Moivre’s Theorem, raise the modulus to the power of n, and multiply the argument by n, applying the trigonometric functions to the result.

## What is De Moivre’s Theorem?

De Moivre’s Theorem is a fundamental formula in complex number theory that allows for the computation of powers and roots of complex numbers. Named after Abraham de Moivre, the theorem states that for any complex number (expressed in polar form) and any integer n, the power of the complex number can be calculated by raising the modulus to the power of n and multiplying the argument by n. This theorem simplifies complex number calculations, especially when dealing with powers and roots.

## How to Use De Moivre’s Theorem?

The following steps outline how to use De Moivre’s Theorem to calculate the power of a complex number:

- Express the complex number in polar form: r(cos θ + i sin θ).
- Determine the modulus (r) and the argument (θ) of the complex number.
- Choose the exponent (n) you wish to raise the complex number to.
- Apply De Moivre’s formula: (r(cos θ + i sin θ)) ^n = r^n (cos(nθ) + i sin(nθ)).
- Calculate r^n and nθ, then use the trigonometric identities to find the result.

**Example Problem:**

Use the following variables as an example problem to test your understanding.

Complex Number = 2(cos π/3 + i sin π/3)

Exponent = 3

## FAQ

**1. What is a complex number?**

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit satisfying i^2 = -1.

**2. What is the polar form of a complex number?**

The polar form of a complex number is a way of expressing the number using its modulus and argument, represented as r(cos θ + i sin θ).

**3. How is De Moivre’s Theorem useful?**

De Moivre’s Theorem is useful for simplifying the calculation of powers and roots of complex numbers, making it easier to work with these numbers in various mathematical and engineering applications.

**4. Can De Moivre’s Theorem be used for roots of complex numbers?**

Yes, De Moivre’s Theorem can be extended to calculate the roots of complex numbers by using fractional exponents.

**5. Is the calculator accurate?**

The calculator provides an estimate based on the inputs provided. For exact figures, it is recommended to verify the results using manual calculations or other mathematical tools.