De Moivre

Definition

De Moivre's Theorem states that for any real number $\theta$ and integer $n$, $(\cos(\theta) + i \sin(\theta))^n = \cos(n\theta) + i \sin(n\theta)$. It is used to raise complex numbers in polar form to a power.

5 Must Know Facts For Your Next Test

1. The theorem can be written as $(r(\cos(\theta) + i \sin(\theta)))^n = r^n (\cos(n\theta) + i \sin(n\theta))$ for a complex number $z = r(\cos(\theta) + i \sin(\theta))$.
2. It simplifies the process of finding powers and roots of complex numbers.
3. The polar form of a complex number is crucial when applying De Moivre's Theorem.
4. The angle $n\theta$ must be adjusted if it's not within the standard interval $[0, 2\pi)$ or $[-\pi, \pi)$, depending on context.
5. De Moivre's Theorem also applies to negative integers, allowing the calculation of roots.

Review Questions

• What is De Moivre's Theorem and how is it written?
• How do you apply De Moivre's Theorem to find the $n$th power of a complex number?
• What adjustment might you need to make to the angle $n\θheta$ when using De Moivre's Theorem?

Related terms

Polar Form: A way of representing a complex number in terms of its magnitude and angle: $z = r(\cos(θ) + i \sin(\θheta))$.

Complex Number: A number of the form $a + bi$, where $i$ is the imaginary unit satisfying $i^2 = -1$.

Euler's Formula: $e^{i\θheta} = \cos(\θheta) + i \sin(\θheta)$, a fundamental formula establishing a deep relationship between exponential functions and trigonometric functions.