4 In an Argand diagram, the complex numbers \(0 , z\) and \(z \mathrm { e } ^ { \frac { 1 } { 6 } \mathrm { i } \pi }\) are represented by the points \(O , A\) and \(B\) respectively.
- Sketch a possible Argand diagram showing the triangle \(O A B\). Show that the triangle is isosceles and state the size of angle \(A O B\).
The complex numbers \(1 + \mathrm { i }\) and \(5 + 2 \mathrm { i }\) are represented by the points \(C\) and \(D\) respectively. The complex number \(w\) is represented by the point \(E\), such that \(C D = C E\) and angle \(D C E = \frac { 1 } { 6 } \pi\).
- Calculate the possible values of \(w\), giving your answers exactly in the form \(a + b \mathrm { i }\).