Polar representation of complex Numbers

Polar coordinate in the plan

Let us consider a coordinate plane and a point M(x,y) that is not the origin.
  => r^2 = x^2 +y^2 
The real number r is called the polar radius  of the M. The direct angel t*€[0, 2pi) between the vector OM and the positive X-axis is called the polar argument of the points M. The pair (r ,t* ) is called the polar coordinate of the point M. 
   The origin O is the unique point such that r=0 ;the argument t* of the origin is not defined. 
   For any point M in the plane there is a unique intersection point P of the ray (OM with the unit circle centered at the origin. The point P has the same polar argument t* using the definition of the sine and cosine functions we find that

    x=rcost* and y=rsint*.

a)If x is not equal 0,from tant*=y/x we deduce that

       t*=arctan(y/x)+k(Pi)

Where
      When x>0 and y>=0 , k=0
      When x<0 and and any y, k=1
      When x>0 and y<0 ,k=2.
b)If x=0 and y is not equal 0,then
    t*=Pi/2 for y>0
    t*=3Pi/2 for y<0

    polar coordinate =M(r, t*)