Quick questions on Polar and exponential form of complex numbers explained: H2 Mathematics Vectors and Complex Numbers

Euler's relation $e^{i\theta} = \cos\theta + i\sin\theta$ gives the compact exponential form:

De Moivre's theorem also runs in reverse to find roots. The $n$ distinct $n$th roots of $re^{i\theta}$ have modulus $r^{1/n}$ and arguments $\tfrac{\theta + 2k\pi}{n}$ for $k = 0, 1, \ldots, n-1$, because adding a full turn of $2\pi$ to the argument before dividing produces a genuinely different root. So the cube roots of $8e^{i\pi}$ have modulus $8^{1/3} = 2$ and arguments $\tfrac{\pi}{3}, \pi, \tfrac{5\pi}{3}$. Geometrically the $n$ roots are equally spaced around a circle of radius $r^{1/n}$, separated by $\tfrac{2\pi}{n}$.

Expanding $(\cos\theta + i\sin\theta)^n$ by de Moivre and comparing real and imaginary parts produces multiple-angle identities, a classic H2 application. For $n = 2$, de Moivre gives $\cos 2\theta + i\sin 2\theta = (\cos\theta + i\sin\theta)^2 = \cos^2\theta - \sin^2\theta + 2i\sin\theta\cos\theta$. Equating real parts yields $\cos 2\theta = \cos^2\theta - \sin^2\theta$ and imaginary parts yields $\sin 2\theta = 2\sin\theta\cos\theta$. Using de Moivre as a generator of trigonometric identities, by expanding and matching parts, connects the complex-number work directly to trigonometry and is a frequently examined technique.

Reduce to $-\pi < \theta \leq \pi$; an argument of $\frac{3\pi}{2}$ should be written $-\frac{\pi}{2}$.

Find the modulus and argument of $z = -1 + i$. [3 marks]

Given $z = 4e^{i\pi/3}$, find $z^2$ in exponential form. [2 marks]

State the rule for the argument of a product of two complex numbers. [1 mark]