4.02s Roots of unity: geometric applications

11 The complex number \(w = ( \sqrt { 3 } - 1 ) + \mathrm { i } ( \sqrt { 3 } + 1 )\).

1. Determine, showing full working, the exact values of \(| w |\) and \(\arg w\).
   [0pt] [You may use the result that \(\tan \left( \frac { 5 } { 12 } \pi \right) = 2 + \sqrt { 3 }\).]
2. (a) Find, in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), the three roots, \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\), of the equation \(z ^ { 3 } = w\).
   (b) Determine \(z _ { 1 } z _ { 2 } z _ { 3 }\) in the form \(a + \mathrm { i } b\).
   (c) Mark the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) on a sketch of the Argand diagram. Show that they form an equilateral triangle, \(\Delta _ { 1 }\), and determine the side-length of \(\Delta _ { 1 }\).
   (d) The points representing \(k z _ { 1 } , k z _ { 2 }\) and \(k z _ { 3 }\) form \(\Delta _ { 2 }\), an equilateral triangle which is congruent to \(\Delta _ { 1 }\), and one of whose vertices lies on the positive real axis. Write down a suitable value for the complex constant \(k\).

Write down the fifth roots of unity. Hence, or otherwise, find all the roots of the equation $$z^5 = -16 + (16\sqrt{3})i,$$ giving each root in the form \(re^{i\theta}\). [4]