# Thread: [SOLVED] Argument of complex number

1. ## [SOLVED] Argument of complex number

Hi !

Sorry if it isn't the correct subforum...

The question is :
Find the argument of $\displaystyle \frac{(z_1+z_2)^2}{z_1z_2}$, given that $\displaystyle |z_1|=|z_2|=1$

I found it to be $\displaystyle 2arg(z_1+z_2)-arg(z_1)-arg(z_2)$, but couldn't go any further.. Did I miss something ?

2. Hi
Originally Posted by Moo
[...] given that $\displaystyle |z_1|=|z_2|=1$
This gives us $\displaystyle z_1=\exp(i\theta)$ and $\displaystyle z_2=\exp(i\varphi)$.

3. Originally Posted by flyingsquirrel
Hi

This gives us $\displaystyle z_1=\exp(i\theta)$ and $\displaystyle z_2=\exp(i\varphi)$.
I know, but well, the z_1+z_2 is disturbing...

I got to expressions like $\displaystyle \cos \frac{\theta_1-\theta_2}{2}$ but couldn't simplify :/

4. Originally Posted by Moo
I know, but well, the z_1+z_2 is disturbing...

I got to expressions like $\displaystyle \cos \frac{\theta_1-\theta_2}{2}$ but couldn't simplify :/
$\displaystyle \frac{(z_1+z_2)^2}{z_1z_2}=\frac{\mathrm{e}^{i2\th eta_1}+2\mathrm{e}^{i(\theta_1+\theta2)}+\mathrm{e }^{i2\theta_2}}{\mathrm{e}^{ i(\theta_1+\theta_2)}}=\mathrm{e}^{i(\theta_1-\theta_2)}+2+\mathrm{e}^{i(\theta_2-\theta_1)}$

If this is the expression you got to, you have the answer since this is a real number.

5. Originally Posted by flyingsquirrel
$\displaystyle \frac{(z_1+z_2)^2}{z_1z_2}=\frac{\mathrm{e}^{i2\th eta_1}+2\mathrm{e}^{i(\theta_1+\theta2)}+\mathrm{e }^{i2\theta_2}}{\mathrm{e}^{ i(\theta_1+\theta_2)}}=\mathrm{e}^{i(\theta_1-\theta_2)}+2+\mathrm{e}^{i(\theta_2-\theta_1)}$

If this is the expression you got to, you have the answer since this is a real number.
I got it, but what I need is the argument... ?
I'm not sure I'm getting what you're saying :/

6. Originally Posted by Moo
I got it, but what I need is the argument... ?
I'm not sure I'm getting what you're saying :/
I should have added that $\displaystyle \frac{(z_1+z_2)^2}{z_1z_2}=\mathrm{e}^{i(\theta_1-\theta_2)}+2+\mathrm{e}^{i(\theta_2-\theta_1)}=2\cos(\theta_1-\theta_2)+2\geq0$

This tells us that the argument is $\displaystyle 0\mod 2\pi$ except when $\displaystyle z_1=-z_2$.

7. Originally Posted by flyingsquirrel
I should have added that $\displaystyle \frac{(z_1+z_2)^2}{z_1z_2}=\mathrm{e}^{i(\theta_1-\theta_2)}+2+\mathrm{e}^{i(\theta_2-\theta_1)}=2\cos(\theta_1-\theta_2)+2\geq0$

This tells us that the argument is $\displaystyle 0\mod 2\pi$ except when $\displaystyle \theta_1=-\theta_2$.
Oh yeah, that's better

Hmm $\displaystyle \theta_1=-\theta_2$ ? ^^

I'd say "arg=0 mod 2pi and theta_1-theta_2=-pi mod 2pi"

Edit : it's ok now

8. Hey Moo

I just have to post the geometric approach to this. hope you like my handwriting!

Bobak

Edit: I wrote equilateral when I meant isosceles, Thanks for spotting that Moo.

9. Nice handwriting !

(should be isoscele though lol)

Thank you both of you !