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 numbers $z_1 = 1 + \sqrt{3}\,i$ and $z_2 = -1 + i$ .
(a) Express $z_1$ and $z_2$ in modulus–argument form, with arguments in $(-\pi, \pi]$ . (3 marks)
(b) Hence find the modulus and argument of $z_1 z_2$ , giving the argument in $(-\pi, \pi]$ . (2 marks)
(c) Hence find the modulus and argument of $\dfrac{z_1}{z_2}$ . (2 marks)
Level 5 · 7 marksTry it
2
Let $z_{1} = 1 + i$ and $z_{2} = \sqrt{3} + i$ .
(a) Express $z_{1}$ and $z_{2}$ in polar form. (4 marks)
(b) Write down the modulus and argument of $z_{1}z_{2}$ . (2 marks)
(c) By also expanding $z_{1}z_{2}$ in Cartesian form, show that

$\cos\frac{5\pi}{12} = \frac{\sqrt{3} - 1}{2\sqrt{2}}$ (2 marks)
Level 5 · 8 marksTry it
3
The complex number $z = -1 + \sqrt{3}\,i$ .
(a) Express $z$ in polar form. (3 marks)
(b) Find $z^{2}$ in the form $a + bi$ , giving $a$ and $b$ exactly. (3 marks)
(c) Show that $z^{3}$ is real, and state its value. (2 marks)
Level 5 · 8 marksTry it