# Find the argument of the complex number.

Tags:
1. Aug 15, 2016

### david102

1. The problem statement, all variables and given/known data
If modulus of z=x+ iy(a complex number) is 1 I.e |z|=1 then find the argument of z/(1+z)^2

2. Relevant equations
argument of z = tan inverse (y/x) where z=x+iy modulus of z is |z|=root(x^2+y^2)

3. The attempt at a solution
z/(1+2z+z^2) = x+iy / 1+2(x+iy)+( x+iy)2 ...........

2. Aug 15, 2016

### Ray Vickson

This is not correct; you have written
$$z/(1+2z+z^2) = x + \frac{iy}{1} + 2(x+iy) + (x+iy)2$$
Did you mean
$$\frac{x + iy}{1 + 2(x+iy) + (x+iy)^2}?$$
If so, use parentheses, like this: (x + iy)/( x + iy + 2(x+iy) + (x+iy)^2). In fact, you need to also expand out (x+iy)^2 to find its real and imaginary parts. Then you need to keep going to find the real and imaginary parts of the entire expression. It will be messy and long, but that's just how it goes sometimes.

Actually, there is another approach that leads to an answer in a couple of lines of simple algebra, but PF rules forbid me from spelling it out.

Last edited: Aug 15, 2016