How Do You Find the Argument of the Sum of Two Complex Numbers?

[Mentallic]

Homework Statement

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

Homework Equations

$$z=x+iy=re^{i\theta}=rcis\theta$$

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 $arg(z_1+z_2)=(\theta+\phi)/2$

 

[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)]

x=radius*cos (angle)
y=radius*sin(angle)

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)

 

[Mark44]

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...

 

[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 $x=tan(3\pi/8)$.