Products and Roots of Complex Numbers in Trigonometric Form

Products and Roots of Complex Numbers

Multiplication and Division of Complex Numbers in Trigonometric Form

Complex numbers can be expressed in trigonometric (polar) form as:

• Trigonometric Form: $z = r (\cos \theta + i \sin \theta)$, where $r$ is the modulus and $\theta$ is the argument.

Multiplying and dividing complex numbers in this form uses the following properties:

• Product: If $z_1 = r_1 (\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2 (\cos \theta_2 + i \sin \theta_2)$, then:

$z_1 z_2 = r_1 r_2 [\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)]$

• Quotient: $\dfrac{z_1}{z_2} = \dfrac{r_1}{r_2} [\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)]$

Example: Multiply $z_1 = 2 (\cos 30^\circ + i \sin 30^\circ)$ and $z_2 = 3 (\cos 45^\circ + i \sin 45^\circ)$:

$z_1 z_2 = 2 \times 3 [\cos(30^\circ + 45^\circ) + i \sin(30^\circ + 45^\circ)] = 6 [\cos 75^\circ + i \sin 75^\circ]$

Roots of Complex Numbers in Trigonometric Form

The $n$th roots of a complex number $z = r (\cos \theta + i \sin \theta)$ are given by:

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

• Each value of $k$ gives a distinct $n$th root, spaced evenly around the unit circle.

Example: Find the cube roots of $8 (\cos 60^\circ + i \sin 60^\circ)$:

$w_k = 2 [\cos(\frac{60^\circ + 360^\circ k}{3}) + i \sin(\frac{60^\circ + 360^\circ k}{3})]$, for $k = 0, 1, 2$

• For $k=0$: $w_0 = 2 [\cos 20^\circ + i \sin 20^\circ]$

• For $k=1$: $w_1 = 2 [\cos 140^\circ + i \sin 140^\circ]$

• For $k=2$: $w_2 = 2 [\cos 260^\circ + i \sin 260^\circ]$

Additional info: These properties are essential for solving equations involving complex numbers and for understanding their geometric representation in the complex plane.