Modulus–Argument (Polar) Form

Modulus–argument (polar) form is a key part of the Complex numbers topic in NSW Year 12 Mathematics Extension 2 (NESA outcome ME2-12-03). This subtopic writes a complex number in modulus–argument form \(z = r(\cos\theta + i\sin\theta)\), where \(r\) is the modulus and \(\theta\) is an argument, and defines the principal argument \(\operatorname{Arg}(z)\) in the interval \((-\pi, \pi]\).

Students and teachers will cover how to convert between Cartesian and polar form, interpret multiplication and division geometrically, and prove and use identities such as \(|z_1 z_2| = |z_1||z_2|\) and \(\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)\) — the language of complex numbers used throughout the HSC Extension 2 course.