3 In this question, \(w\) denotes the complex number \(\cos \frac { 2 } { 5 } \pi + \mathrm { i } \sin \frac { 2 } { 5 } \pi\).
- Express \(w ^ { 2 } , w ^ { 3 }\) and \(w ^ { * }\) in polar form, with arguments in the interval \(0 \leqslant \theta < 2 \pi\).
- The points in an Argand diagram which represent the numbers
$$1 , \quad 1 + w , \quad 1 + w + w ^ { 2 } , \quad 1 + w + w ^ { 2 } + w ^ { 3 } , \quad 1 + w + w ^ { 2 } + w ^ { 3 } + w ^ { 4 }$$
are denoted by \(A , B , C , D , E\) respectively. Sketch the Argand diagram to show these points and join them in the order stated. (Your diagram need not be exactly to scale, but it should show the important features.)
- Write down a polynomial equation of degree 5 which is satisfied by \(w\).