Complex numbers confused!!

Question

If you give me a complex number say $z=2+3i$, then I can easily find $\text{Im}(z)=3$ and $\text{Re}(z)=2$ but when this polar coordinates stuff came, I lost my head!

So say $z=r(\cos\theta+i\sin\theta)$. What is $\text{Im}(z)$ and $\text{Re}(z)$ ?

Answer 1

Just expand and you'll get: $$z=\underbrace{r\cos\theta}_{\displaystyle\Re(z)}+i\underbrace{\,r\sin\theta}_{\displaystyle\Im(z)}.$$

I hope this helps.
Best wishes, $\mathcal H$akim.

Answer 2

$r[\cos(\theta)] = \operatorname{Re}(z)$, $r[\sin(\theta)] = \operatorname{Im}(z)$