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
Using $z_1 = 2\left(\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}\right)$ and $z_2 = \sqrt{2}\left(\cos\dfrac{3\pi}{4} + i\sin\dfrac{3\pi}{4}\right)$ , express $\dfrac{z_1}{z_2}$ in the form $r(\cos\theta + i\sin\theta)$ , where $r > 0$ and $-\pi < \theta \le \pi$ . (3 marks)
Level 4 · 3 marksTry it
2
The complex number $z = 3\left(\cos\dfrac{\pi}{5} + i\sin\dfrac{\pi}{5}\right)$ .
(a) Write down $z^{2}$ in polar form. (2 marks)
(b) Find the smallest positive integer $n$ for which $z^{n}$ is real, and state the value of $z^{n}$ for this $n$ . (3 marks)
Level 4 · 5 marksTry it