8

8. [Maximum mark: 10]

Let $z = 1 + \cos(2\theta) + i \sin(2\theta)$, where $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$.

(a) Show that
(i) $\arg z = \theta$; [7]

(ii) $|z| = 2 \cos \theta$. [7]

(b) Hence or otherwise, find the value of $\theta$ such that $\arg(z^2) = |z^3|$. [3]