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 $z = 1 + \sqrt{3}\,i$ .
(a) Find $|z|$ . (1 mark)
(b) Find $\arg z$ , and hence express $z$ in the form $r(\cos\theta + i\sin\theta)$ , where $-\pi < \theta \le \pi$ . (2 marks)
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