Complex number - arctan $$z=-2+2\sqrt{3}i\implies x=-2, y=2\sqrt3$$
$$r=\sqrt{x^2+y^2}=\sqrt{4+12}=4$$
$$\text{Angle}=\arctan\left(\frac{2\sqrt3}{-2}\right)+\pi=\frac{2\pi}{3}=120^\circ$$
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! 


Notice, $\tan(-\theta)=-\tan\theta$
$$\therefore \arctan\left(\frac{2\sqrt3}{-2}\right)+\pi=\arctan\left(-\sqrt{3}\right)+\pi=-\frac{\pi}{3}+\pi=\frac{2\pi}{3}=120^\circ$$
You can use calculator to find amplitude & argument. Do remember $\tan^{-1}\sqrt3=\frac{\pi}{3}$ 
