formula for argument of a complex number

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

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