De Moivre’s Theorem

Definition

De Moivre's Theorem states that for any complex number in polar form $r(\cos \theta + i\sin \theta)$ and integer $n$, $(r(\cos \theta + i\sin \theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))$. It provides a way to raise complex numbers to powers and extract roots using trigonometric functions.

5 Must Know Facts For Your Next Test

1. De Moivre's Theorem is used to simplify the process of raising complex numbers to a power.
2. The theorem extends to finding nth roots of complex numbers by manipulating angles and magnitudes.
3. It is crucial for converting between rectangular and polar forms of complex numbers.
4. The angle $\theta$ in De Moivre's Theorem is measured in radians.
5. De Moivre's Theorem can be derived from Euler's formula: $e^{i\theta} = \cos \theta + i\sin \theta$.

Review Questions

• What is De Moivre's Theorem, and how does it simplify the calculation of powers of complex numbers?
• How would you use De Moivre’s Theorem to find the cube of a complex number given in polar form?
• Explain how De Moivre’s Theorem assists in finding the nth roots of complex numbers.

Related terms

Polar Form: A way to represent complex numbers as $r(\cos \theta + i\sin \theta)$, where $r$ is the magnitude and $θ$ is the angle.

Euler’s Formula: $e^{i\theta} = \cos \theta + i\sin \theta$, which connects exponential functions with trigonometric functions.

Magnitude (Modulus): The distance from the origin to the point representing the complex number on the complex plane, denoted as $|z|$.