# How to find the range of a complex number's argument?

If $\arg(z-2-2\mathrm{i}) = 60$ then find the range of $\arg(z)$.

I don't know how to find the range of a complex number.

• It basically tells you to find alll possible complex numbers $z$ such that $z-2-2i$ lies on a specific line (technically a ray). Finding the range of the argument of $z$ is secondary, and is easily adressed once you find all the relevant $z$. I would personally recommend you solve this problem geometrically, by drawing in the complex plane. – Arthur Jun 12 '17 at 11:31

If $z=x+iy$ where $x,y$ are real using atan2
$x-2>0\iff x>2\ \ \ \ (1)$
$y-2\ge0\iff y\ge2\ \ \ \ (2)$
So, $arg(x-2+iy-2i)=60^\circ\implies\dfrac{y-2}{x-2}=\sqrt3$ honoring $(1),(2)$