Argument of a complex \[z=a+ib\] no is defined as the angle subtended by the vector z from origin.
It varies from -\[\Pi , +\Pi \] in radians or -180 to +180 in degrees.
It is different for different quadrants
Case 1 - when z is in 1st quadrant
\[z=a+ib\] a> 0 , b> 0
arg(z)=\[ tan^{-1}(\frac{b}{a})\]
for example z=\[ 2+2i\]
arg(z)=\[tan^{-1}(\frac{2}{2})=45^0\] here a=2 and b=2
case2- when Z is in 2nd quadrant
Z=a+ib a < 0 and b > 0
arg(Z)= \[ \Pi +tan^{-1}(\frac{b}{a})\]
Example z=- 1+\[ \sqrt{3}i\]
here a=-1 and b= \[ \sqrt{3}\]
hence argument is given by arg(Z)= \[ \Pi +tan^{-1}(\frac{\ \sqrt{3}}{-2})\]
=\[\Pi -\frac{\Pi }{3}=\frac{2\Pi }{3}\] radians = 120 degrees.
Case3- When z is in 3rd quadrant.
Z= a+ib a < 0, b <0
arg(z)= \[- \Pi +tan^{-1}(\frac{b}{a})\]
Example- Let Z= -2-2i
here a=-2 and b=-2
So arg(z)= \[- \Pi +tan^{-1}(\frac{-1}{-1})\]
=\[ -\frac{3\Pi }{4}\] radians= -135 degrees
Case 4 When z is in 4th quadrant .
Z=a+ib where a > 0 and b < 0
arg(z)=\[ tan^{-1}(\frac{b}{a})\]
for example z=\[ 2-2i\]
arg(z)=\[tan^{-1}(\frac{2}{-2})=-45^0\] here a=2 and b=-2