# Complex number arguments

1. Feb 17, 2010

### Mentallic

1. The problem statement, all variables and given/known data
If given two complex numbers z1 and z2 that have arguments $\theta$ and $\phi$, and moduli r and R respectively, then find an expression for the mod-arg form of z1+z2

2. Relevant equations
$$z=x+iy=re^{i\theta}=rcis\theta$$

3. The attempt at a solution
I can't seem to find a way to relate z1+z2 since I would need to somehow combine the trigonometry terms of:
$$z_1+z_2=rcis\theta+Rcis\phi=rcos\theta+Rcos\phi+i(rsin\theta+Rsin\phi)$$

In a similar fashion, $$z_1z_2=rcis\theta.Rcis\phi=rRcis(\theta+\phi)$$ which does have a relationship. Can I do anything to that equation to find the argument of the new complex number z1+z2?

I'm aware that I can convert both complex numbers into x+iy form and then go from there and also if there are some simple values for r and R, such as r=R then the $arg(z_1+z_2)=(\theta+\phi)/2$

2. Feb 17, 2010

### korican04

When you add complex numbers it is the same as adding 2d vectors.
so your
radius = sqrt[ (x1+x2)^2 + (y1+y2)^2]
angle = tan-1 [(y1+y2)/(x1+x2)]

After some expansion and trig formulas
r= sqrt[r^2 + R^2 + 2rRcos(theta-phi)]
angle=tan-1[ rsin(theta)+Rsin(phi) / rcos(theta) + Rsin(phi)

3. Feb 17, 2010

### Staff: Mentor

Seems like it would be a lot easier to convert the two complex numbers to rectangular form, and then add components, then convert the sum back to polar form. It might be that's what korican04 was saying...

4. Feb 17, 2010

### Mentallic

Korican04, the final result you obtain doesn't help my situation. Thanks for the attempt though

Mark, the reason why I wanted to avoid converting to rectangular form is because when I was helping someone with a question which required to find arg(z1+z2) where z1 and z2 were some known complex numbers which I don't remember right now, the answer turned out to be $3\pi/8$ and both me and the guy I was helping haven't been taught to easily recognize the number $x=tan(3\pi/8)$.