NLed, Sorry to be so late with the following answer. (I found your posting because I am looking for software to construct an Argand diagram just like the one in the gray area in the original posting and need help. I hope you will contact me at onekevATgmailDOTcom.)
The decision to add 180 degrees to the inverse tangent is based on the sign of the denominator "inside" the inverse tangent. Add 180 degrees only if denominator < 0. Also, for this procedure to work, use angles (in degrees) in (-180, 180] instead of the more typical [0, 360).
The angle interval (-180, 180] is more commonly used in the study of functions of a complex variable and electrical engineering than the (perhaps more intuitive) interval [0, 360) used in trigonometry and pre-calculus.
But if you prefer an angle in [0, 360), use the scheme described above then add 360 degrees only if the angle value found is negative.