4 The complex number \(u\) is given by \(u = - 1 - \mathrm { i } \sqrt { 3 }\).
- 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 \(v = 5 \left( \cos \frac { 1 } { 6 } \pi + \mathrm { i } \sin \frac { 1 } { 6 } \pi \right)\). - Express the complex number \(\frac { \mathrm { v } } { \mathrm { u } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).