Showing your working, find the two square roots of the complex number \(1 - ( 2 \sqrt { } 6 ) \mathrm { i }\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact.
On a sketch of an Argand diagram, shade the region whose points represent the complex numbers \(z\) which satisfy the inequality \(| z - 3 i | \leqslant 2\). Find the greatest value of \(\arg z\) for points in this region.