$\displaystyle \frac{2i}{2+i}$
i don know how to separate the complex and imaginary part of this fracture?
$\displaystyle \frac{2i}{2 + i} = \frac{2i(2 - i)}{(2 + i)(2 - i)}$
$\displaystyle = \frac{4i - 2i^2}{4 - i^2}$
$\displaystyle = \frac{2 + 4i}{5}$
$\displaystyle = \frac{2}{5} + \frac{4}{5}i$.
Now putting into polar form...
$\displaystyle |z| = \sqrt{\left(\frac{2}{5}\right)^2 + \left(\frac{4}{5}\right)^2}$
$\displaystyle = \sqrt{\frac{4}{25} + \frac{16}{25}}$
$\displaystyle = \sqrt{\frac{20}{25}}$
$\displaystyle = \frac{2\sqrt{5}}{5}$.
$\displaystyle \theta = \arctan{\frac{\frac{4}{5}}{\frac{2}{5}}}$
$\displaystyle = \arctan{2}$.
So $\displaystyle z = \frac{2\sqrt{5}}{5}\,\textrm{cis}\,\arctan{2}$
$\displaystyle = \frac{2\sqrt{5}}{5}e^{i\arctan{2}}$.