I've to sketch the complex numbers $z$ satisfying both the inequalities
$$|(z-2i)|\le2,$$
$$ 0\le \arg(z+2)\le 45^\circ.$$
I was able to sketch and shade the region that satisfies both inequalities; here is my Argand diagram:
However, I've a problem in getting the greatest value of $|z|$, i.e. the maximum length of $z$ from $(0,0).$ It should come to around 3.70.
Best Answer
if you parametrize the circle of constraint it is $(2\cos t,2\sin t +2)$ with $t=0$ corresponding to the point $(2,2)$. so:
$$ \frac18 |z|^2 = \frac12 (\cos^2 t +\sin^2 t + 2 \sin t +1 )= 1+ \sin t $$