- Plot complex numbers in the complex plane.
- Write complex numbers in polar form.
- Convert a complex number from polar to rectangular form.
- Find products and quotients of complex numbers in polar form.
- Find powers and roots of complex numbers in polar form.
De Moivre’s Theorem for Powers
If [latex]z = r(\cos\theta + i\sin\theta)[/latex], then:
[latex]z^n = r^n(\cos(n\theta) + i\sin(n\theta))[/latex]
This theorem also works in reverse to find nth roots of complex numbers, which helps us solve equations like [latex]x^n = c[/latex] where [latex]c[/latex] is any complex number.
Solve [latex]x^3=8[/latex]
Solve the equation [latex]x^5 = 32[/latex] by finding all five complex roots.