Let&nbsp;\(z = r(\cos \theta + i \sin \theta)\) and \(n \in \mathbb{Z}\). Then
\[z^n = r^n[\cos(n\theta) + i \sin(n\theta)].\]
In particular, when&nbsp;\(r = 1\), one has
\[(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta).\]

Let \(z = r(\cos \theta + i \sin \theta)\) and \(n \in \mathbb{Z}\). Then

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

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

Properties from Exponential or Trigonometric form [Out of Syllabus]

Complex Numbers

International Baccalaureate Higher Level Mathematics

De Moivre's Theorem

De Moivre's Theorem