I am stuck on a past exam question. I don't have a clue what it's on about and would appreciate any help.
In this question, $w$ denotes the complex number $cos{\frac{2}{5}\pi} + isin{\frac{2}{5}\pi}$
i) Express $w^2$, $w^3$ and $w^*$ in polar form, with arguments in the interval $0 \le \theta < 2\pi$
My working (I'm pretty sure this is right just it's part of the same question) $|w| = 1$ $arg(w) = \frac{2}{5}\pi$ $\therefore |w^2| = 1$ $arg(w^2) = \frac{2}{5}\pi + \frac{2}{5}\pi = \frac{4}{5}\pi$ $|w^3| = 1$ $arg(w^3) = \frac{2}{5}\pi + \frac{2}{5}\pi + \frac{2}{5}\pi = \frac{6}{5}\pi$ $\therefore w^2 = cos{\frac{4}{5}\pi} + isin{\frac{4}{5}\pi}, w^3 = cos{\frac{6}{5}\pi} + isin{\frac{6}{5}\pi}, w^* = cos{\frac{2}{5}\pi} - isin{\frac{2}{5}\pi}$
ii) The points in an Argand Diagram which represent the number $ 1, 1 + w, 1 + w + w^2, 1 + w + w^2 + w^3, 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.
I could do this by brute force just bashing numbers into my calculator to find the coordinates but surely there has to be a better way? Is there a property of complex numbers when they're added together like that?