A-level · Complex numbers

Modulus–argument (polar) formExam-style questions

Writing z = r(cos θ + i sin θ) with r = |z| and θ = arg z (the principal value in (−π, π]), converting between Cartesian a + bi and polar form both ways, and combining numbers in polar form: to MULTIPLY you multiply the moduli and ADD the arguments, to DIVIDE you divide the moduli and SUBTRACT the arguments — then adjust the resulting argument back into the principal range.

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16 questions · A Level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

1
The complex number $u$ is given by $u = 3 - 3\sqrt{3}\,\mathrm{i}$ .
(a) Express $u$ in the form $r(\cos\theta + \mathrm{i}\sin\theta)$ , where $r > 0$ and $-\pi < \theta \leqslant \pi$ . Give the exact values of $r$ and $\theta$ .

(2 marks)

The complex number $v$ is given by $v = 2\left(\cos\tfrac{3}{4}\pi + \mathrm{i}\sin\tfrac{3}{4}\pi\right)$ .
(b) Express the complex number $\dfrac{u}{v}$ in the form $r\mathrm{e}^{\mathrm{i}\theta}$ , where $r > 0$ and $-\pi < \theta \leqslant \pi$ .

(2 marks)
Level 3 · 4 marksTry it
2
Given $z_1 = 2\left(\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}\right)$ and $z_2 = \sqrt{2}\left(\cos\dfrac{3\pi}{4} + i\sin\dfrac{3\pi}{4}\right)$ , express $z_1 z_2$ in the form $r(\cos\theta + i\sin\theta)$ , where $r > 0$ and $-\pi < \theta \le \pi$ . (3 marks)
Level 3 · 3 marksTry it
3
Express $z = 6\left(\cos\dfrac{5\pi}{6} + i\sin\dfrac{5\pi}{6}\right)$ in the form $a + bi$ , giving $a$ and $b$ exactly. (3 marks)
Level 3 · 3 marksTry it
4
Given

$z_{1} = 8\left(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4}\right), \qquad z_{2} = 2\left(\cos\frac{\pi}{12} + i\sin\frac{\pi}{12}\right)$

find, in the form $r(\cos\theta + i\sin\theta)$ :
(a) $z_{1}z_{2}$ (2 marks)
(b) $\dfrac{z_{1}}{z_{2}}$ (2 marks)
Level 3 · 4 marksTry it
5
The complex number $z = \sqrt{2}\left(\cos\dfrac{\pi}{8} + i\sin\dfrac{\pi}{8}\right)$ .

Find $z^{2}$ in the form $a + bi$ , giving $a$ and $b$ exactly. (4 marks)
Level 3 · 4 marksTry it

2 more questions on modulus–argument (polar) form

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