What does the De Moivre's Theorem say?

De Moivre's Theorem provides a formula for computing integer powers of complex numbers expressed in polar form. It states that for any real number $\theta$ and integer $n$, $(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)$. When a modulus $r$ is included, the theorem generalizes to $(r(\cos \theta + i \sin \theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))$.

What is an example of the De Moivre's Theorem?

$(\cos(\pi/6) + i\sin(\pi/6))^3 = \cos(3 \cdot \pi/6) + i\sin(3 \cdot \pi/6) = \cos(\pi/2) + i\sin(\pi/2) = i$

Complex NumbersMath Rule

De Moivre's Theorem

Q: What should I remember about the De Moivre's Theorem?

The theorem is valid for all integers $n$, including negative integers. It is a fundamental tool for finding the $n$-th roots of complex numbers.

De Moivre's Theorem provides a formula for computing integer powers of complex numbers expressed in polar form. It states that for any real number $θ$ and integer $n$ , $(cos θ + i sin θ)^{n} = cos (n θ) + i sin (n θ)$ . When a modulus $r$ is included, the theorem generalizes to $(r (cos θ + i sin θ))^{n} = r^{n} (cos (n θ) + i sin (n θ))$ .

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(r (cos θ + i sin θ))^{n} = r^{n} (cos (n θ) + i sin (n θ))

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The rule

Description

See it in action

Examples

(cos (π /6) + i sin (π /6))^{3} = cos (3 \cdot π /6) + i sin (3 \cdot π /6) = cos (π /2) + i sin (π /2) = i

To find $(1 + i)^{4}$ , convert $1 + i$ to polar form: $2 (cos (π /4) + i sin (π /4))$ . Then $(1 + i)^{4} = (2)^{4} (cos (4 π /4) + i sin (4 π /4)) = 4 (cos (π) + i sin (π)) = - 4$ .

Good to know

Key Facts

The theorem is valid for all integers $n$ , including negative integers.
It is a fundamental tool for finding the $n$ -th roots of complex numbers.

Common questions

Frequently Asked Questions

$(cos (π /6) + i sin (π /6))^{3} = cos (3 \cdot π /6) + i sin (3 \cdot π /6) = cos (π /2) + i sin (π /2) = i$

The theorem is valid for all integers $n$ , including negative integers. It is a fundamental tool for finding the $n$ -th roots of complex numbers.