Powers and Roots of Complex Numbers (Trigonometric Form)

Powers and Roots of Complex Numbers (Trigonometric Form)

Introduction to Complex Numbers in Trigonometric Form

Complex numbers can be expressed in trigonometric (polar) form, which is especially useful for finding powers and roots. This approach utilizes the modulus and argument of the complex number, connecting algebraic and geometric perspectives.

• Trigonometric Form: Any complex number $z$ can be written as $z = r(\cos \theta + i \sin \theta)$, where $r$ is the modulus and $\theta$ is the argument.

• Modulus ($r$): The distance from the origin to the point in the complex plane.

• Argument ($\theta$): The angle formed with the positive real axis.

De Moivre's Theorem

De Moivre's Theorem provides a formula for raising complex numbers in trigonometric form to integer powers:

• Theorem Statement:

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

• This theorem simplifies the computation of powers of complex numbers.

Finding Roots of Complex Numbers

The $n$th roots of a complex number can be found using the following formula:

• Formula for Roots:

$\sqrt[n]{r} \left[ \cos \left( \frac{\theta + 2\pi k}{n} \right) + i \sin \left( \frac{\theta + 2\pi k}{n} \right) \right]$, where $k = 0, 1, ..., n-1$

• This formula gives all $n$ distinct roots, spaced evenly around the unit circle.

Example: Finding Roots

Find all cube roots of $z = 2 \text{cis} 255^\circ$.

• Step 1: Identify $r = 2$, $\theta = 255^\circ$.

• Step 2: Use the root formula for $n = 3$:

$z_k = \sqrt[3]{2} \text{cis} \left( \frac{255^\circ + 360^\circ k}{3} \right)$, for $k = 0, 1, 2$

• Step 3: Calculate each root:

k	Angle	Root
0	$\frac{255^\circ}{3} = 85^\circ$	$\sqrt[3]{2} \text{cis} 85^\circ$
1	$\frac{255^\circ + 360^\circ}{3} = \frac{615^\circ}{3} = 205^\circ$	$\sqrt[3]{2} \text{cis} 205^\circ$
2	$\frac{255^\circ + 720^\circ}{3} = \frac{975^\circ}{3} = 325^\circ$	$\sqrt[3]{2} \text{cis} 325^\circ$

How to Find Roots of Complex Numbers (Summary Steps)

1. Find $r$ and $\theta$ for the complex number.

2. Compute $\sqrt[n]{r}$.

3. For each $k = 0, 1, ..., n-1$, calculate $\theta_k = \frac{\theta + 360^\circ k}{n}$ (or $2\pi k$ in radians).

4. Write each root as $\sqrt[n]{r} \text{cis} \theta_k$.

Additional info:

• "cis" notation is shorthand for $\cos \theta + i \sin \theta$.

• Roots of unity are a special case where $r = 1$ and $\theta = 0$.