Suppose \(z=\dfrac{4\sqrt2\,\mathrm{i}}{\sqrt6+\sqrt2\,\mathrm{i}}\), where \(\mathrm{i}^2=-1\).
Show that \(z=1+\sqrt3\,\mathrm{i}\).[2]
Hence, find in terms of \(n\),[4]
\(\arg(z^n)\)
\(|z^n|\)
If \(w^2=z\), find the possible values of \(w\), leaving your answer in the form \(r\mathrm{e}^{\mathrm{i}\theta}\) where \(r>0\) and \(\theta\in(-\pi,\pi]\).[4]
Let \(z^n=r\big(\mathrm{f}(n)+\mathrm{i}\,\mathrm{g}(n)\big)\) for all \(n\in\mathbb Z^+\), where \(\mathrm{f}\) and \(\mathrm{g}\) are trigonometric functions of \(n\) and \(r=|z^n|\).
Find \(\mathrm{f}(1)\), \(\mathrm{f}(2)\) and \(\mathrm{f}(3)\).[4]
Calculate the product of the first \(12\) values of \(\mathrm{f}(n)\), that is, evaluate \[\mathrm{f}(1)\times\mathrm{f}(2)\times\mathrm{f}(3)\times\cdots\times\mathrm{f}(11)\times\mathrm{f}(12).\][3]