$$z=-2+2\sqrt{3}i\implies x=-2, y=2\sqrt3$$ $$r=\sqrt{x^2+y^2}=\sqrt{4+12}=4$$ $$\text{Angle}=\arctan\left(\frac{2\sqrt3}{-2}\right)+\pi=\frac{2\pi}{3}=120^\circ$$
1) May I know how $\arctan\left(\dfrac{2\sqrt3}{-2}\right)+\pi$ turns into $\dfrac{2\pi}{3}$?
2) Can I use calculator to do the calculation and how?
Thanks for the kindness!