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). This subtopic uses proof by mathematical induction to prove de Moivre's theorem for positive integer powers, \((\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta\), and then proves the same result holds for negative integers \(n\) — giving a fast, reliable way to raise a complex number in modulus–argument (polar) form to a power.

Students and teachers will cover how to use de Moivre's theorem to find any integer power of a given complex number and how to derive trigonometric identities such as expansions of \(\cos n\theta\) and \(\sin n\theta\), moving confidently between polar and Cartesian form. These are the powers-of-complex-numbers skills assessed throughout the HSC Extension 2 course.