Without using a calculator, express the complex number \(\frac { 2 + 6 \mathrm { i } } { 1 - 2 \mathrm { i } }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
Hence, without using a calculator, express \(\frac { 2 + 6 \mathrm { i } } { 1 - 2 \mathrm { i } }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).
On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(| z - 3 \mathrm { i } | \leqslant 1\) and \(\operatorname { Re } z \leqslant 0\), where \(\operatorname { Re } z\) denotes the real part of \(z\). Find the greatest value of \(\arg z\) for points in this region, giving your answer in radians correct to 2 decimal places.