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

DifficultyAllLevel 1Level 2Level 3Level 4Level 5
  1. 1
    Express each of the following in the form r(cos⁡θ+isin⁡θ)r(\cos\theta + i\sin\theta), where r>0r > 0 and −π<θ≤π-\pi < \theta \le \pi.
    (a) 3+i\sqrt{3} + i (2 marks)
    (b) −1+i-1 + i (2 marks)
    Level 2 · 4 marksTry it
  2. 2
    The complex number z=4(cos⁡2π3+isin⁡2π3)z = 4\left(\cos\dfrac{2\pi}{3} + i\sin\dfrac{2\pi}{3}\right).

    Express zz in the form a+bia + bi, giving aa and bb exactly. (2 marks)
    Level 2 · 2 marksTry it
  3. 3
    Express each of the following in the form r(cos⁡θ+isin⁡θ)r(\cos\theta + i\sin\theta) with r>0r > 0 and −π<θ⩽π-\pi < \theta \leqslant \pi:
    (a) z=−5iz = -5i (2 marks)
    (b) z=−4z = -4 (2 marks)
    Level 2 · 4 marksTry it