Given $D=\{r(\sin(\alpha)+i\cos(\alpha))\mid r>0\}= \{e^{x+ i y} \mid x\in \mathbb{R} \}$ (for fixed a $y$ and $\alpha$) i need to find $f(D)$ where $f$ is the the joukowski transform: $f(z)=\frac{1}{2}(z+\frac{1}{z})$
After playing around with it i got to ($y=\theta$) $\frac{\sin\left(\theta\right)}{2}\left(e^{x}-e^{-x}\right)+i\frac{\cos\left(\theta\right)}{2}\left(e^{x}+e^{-x}\right)$
I am afraid i got somewhat lost, Any suggestions would be appreciated.