De Moivre's Theorem

De Moivre's Theorem is a fundamental result in complex number theory that connects complex numbers with trigonometry. It is named after the French mathematician Abraham de Moivre.

Statement of De Moivre's Theorem

De Moivre's Theorem states that for any real number $\theta$ and any integer $n$, the following equation holds:

$$(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)$$

Here, $i$ is the imaginary unit, satisfying $i^2 = -1$.

Understanding the Theorem

To understand De Moivre's Theorem, it's important to recognize that a complex number can be represented in polar form as $r(\cos \theta + i \sin \theta)$, where $r$ is the magnitude (or modulus) of the complex number and $\theta$ is the argument (or angle).

When we raise this complex number to an integer power $n$, De Moivre's Theorem provides a straightforward way to calculate the result.

Application of De Moivre's Theorem

De Moivre's Theorem is particularly useful for:

• Computing powers of complex numbers
• Finding the nth roots of complex numbers
• Solving trigonometric identities
• Analyzing oscillations and waves in physics and engineering

Formulas Derived from De Moivre's Theorem

Using De Moivre's Theorem, we can derive formulas for $\cos(n\theta)$ and $\sin(n\theta)$ in terms of powers of $\sin(\theta)$ and $\cos(\theta)$.

For example, for $n = 2$, we have:

$$(\cos \theta + i \sin \theta)^2 = \cos^2 \theta - \sin^2 \theta + 2i \cos \theta \sin \theta$$

Which matches the expressions for $\cos(2\theta)$ and $\sin(2\theta)$:

• $\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$
• $\sin(2\theta) = 2 \cos \theta \sin \theta$

Examples

Let's illustrate De Moivre's Theorem with a couple of examples.

Example 1: Power of a Complex Number

Calculate $(1 + i)^4$ using De Moivre's Theorem.

First, express $1 + i$ in polar form. We find that $r = \sqrt{2}$ and $\theta = \frac{\pi}{4}$. Thus:

$$(1 + i)^4 = (\sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}))^4 = 2^2(\cos(4 \cdot \frac{\pi}{4}) + i \sin(4 \cdot \frac{\pi}{4})) = 4(\cos \pi + i \sin \pi) = -4$$

Example 2: nth Roots of a Complex Number

Find the cube roots of $8(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3})$.

Using De Moivre's Theorem, we know that the nth roots are given by:

$$r^{1/n}(\cos(\frac{\theta + 2k\pi}{n}) + i \sin(\frac{\theta + 2k\pi}{n}))$$

For $n = 3$ and $k = 0, 1, 2$, we get:

• $2(\cos(\frac{\pi}{9}) + i \sin(\frac{\pi}{9}))$
• $2(\cos(\frac{\pi}{9} + \frac{2\pi}{3}) + i \sin(\frac{\pi}{9} + \frac{2\pi}{3}))$
• $2(\cos(\frac{\pi}{9} + \frac{4\pi}{3}) + i \sin(\frac{\pi}{9} + \frac{4\pi}{3}))$

Differences and Important Points

Aspect	Description
Polar Form	De Moivre's Theorem applies to complex numbers in polar form.
Integer Exponents	The theorem is valid for any integer exponent $n$.
Trigonometric Connection	It provides a direct link between complex number exponentiation and trigonometric functions.
Roots of Complex Numbers	The theorem can be used to find the nth roots of complex numbers.
Trigonometric Identities	It can be used to derive trigonometric identities.

Conclusion

De Moivre's Theorem is a powerful tool in the study of complex numbers and trigonometry. It simplifies the process of raising complex numbers to integer powers and finding their roots, and it has applications in various fields of science and engineering. Understanding this theorem is essential for anyone working with complex numbers or trigonometric functions.