Argand Diagram - Quadrants help I forgot the rules of adding angles when it comes to argand diagrams.

In the first quadrant, you add 90 degrees to whatever angle you get, what about Q2 Q3 Q4 ?

This picture will explain what i mean :



notice how 18.43 was added with 180 .. can someone list the rules for the other quadrants as well ?

what if i have -3i -3j for example ?

Thanks
When you measure the argument of a complex number you go counterclockwise from the x-axis.



So to find the argument you take $\arctan\left(\frac{b}{a}\right)$. However the range of the $\arctan x$ function is only from $[\frac{-\pi}{2},\frac{\pi}{2}]$. That is to say, you only get values between $-90^{\circ}$ and $90^{\circ}$ for whatever value you put in $\arctan x$. That means you are completely missing the $2$nd and $3$rd quadrant!

To account for the $2$nd and $3$rd quadrant we must first remember that $\tan \theta$ has period $\pi$. With that we realize that while $\theta+\pi$ has a different value from $\theta$; $\tan(\theta+\pi)=\tan(\theta)$. In fact it turns out $\tan(\theta +k\pi)=\tan(\theta),k\in\mathbb{Z}$. That means there are infinitely many solutions to $\tan x =\theta$ of the form $x=\arctan(\theta)+k\pi,k\in\mathbb{Z}$. But since $2\pi$ is one whole round around a circle all a lot of the solutions overlap and we are left with $2$ unique answers: $$x=\arctan(\theta)$$
$$x=\arctan(\theta)+\pi$$
However, your calculator does not know which solution you want and so it always gives you the first one. To get the second solution you simply add $\pi$ (or $180^{\circ}$ if you like degrees).
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 &lt; 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.