Sketch on an Argand diagram the locus, \(C\), of points for which \(| z - 4 | = 3\).
By drawing appropriate lines through the origin, indicate on your Argand diagram the point A on the locus \(C\) where \(\arg z\) has its maximum value. Indicate also the point B on the locus \(C\) where \(\arg z\) has its minimum value.
Given that \(\arg z = \alpha\) at A and \(\arg z = \beta\) at B , indicate on your Argand diagram the set of points for which \(\beta \leqslant \arg z \leqslant \alpha\) and \(| z - 4 | \geqslant 3\).
Calculate the value of \(\alpha\) and the value of \(\beta\).