I need a little help with this proof. I haven't got any idea how to proceed. The pseudo clue in the question doesn't clarify much.
The real numbers $r$ and $\theta$, where $r > 0$ and $-\pi < \theta < \pi$, are such that, $rcos\theta + 2r^2cos2\theta + 3r^3cos3\theta = 0$ $rsin\theta + 2r^2sin2\theta + 3r^3sin3\theta = 0$ By writing $z = r(cos\theta + isin\theta)$,
Show that $z = \frac{1}{3}(-1 \pm i\sqrt2)$.
Deduce the value of $r$ and the 2 possible values of $tan\theta$.
The only thing I see is the last part of the question. Deducing $r$ and $tan\theta$ from $z = \frac{1}{3}(-1 \pm i\sqrt2)$ would be simple. I calculated $r = \frac{1}{\sqrt3}$ and $tan\theta = \pm\sqrt2$.
Thanks for your help!