Convert to exponential/polar form

A question is this type if and only if it asks to express a given complex number (in Cartesian form) in the form r·e^(iθ) or r(cos θ + i sin θ), typically requiring calculation of modulus and argument.

2 questions · Moderate -0.2

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CAIE P3 2014 June Q5
8 marks Standard +0.3
5 The complex number \(z\) is defined by \(z = \frac { 9 \sqrt { } 3 + 9 i } { \sqrt { } 3 - i }\). Find, showing all your working,
  1. an expression for \(z\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\),
  2. the two square roots of \(z\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
CAIE P3 2024 June Q4
4 marks Moderate -0.8
4 The complex number \(u\) is given by \(u = - 1 - \mathrm { i } \sqrt { 3 }\).
  1. 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 \(\theta\).
    The complex number \(v\) is given by \(v = 5 \left( \cos \frac { 1 } { 6 } \pi + \mathrm { i } \sin \frac { 1 } { 6 } \pi \right)\).
  2. Express the complex number \(\frac { \mathrm { v } } { \mathrm { u } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).