De Moivre's Theorem

De Moivre's theorem is a core part of complex numbers in NSW Year 12 Mathematics Extension 2 (NESA outcome ME2-12-03). The theorem states \((\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta\). Students prove it by mathematical induction when \(n\) is a positive integer, then extend it to negative integers — giving a fast, reliable way to raise a complex number in polar form to a power.

Students and teachers will cover evaluating powers such as \((1+i)^5\) or \((1+\sqrt{3}\,i)^3\) by switching to polar form, and expanding \((\cos\theta + i\sin\theta)^n\) with the binomial theorem to obtain multiple-angle identities for \(\cos n\theta\) and \(\sin n\theta\) in powers of \(\cos\theta\) and \(\sin\theta\) — skills that appear throughout the HSC Extension 2 course.