# Thread: Complex number question, involving argument and modulus

1. ## Complex number question, involving argument and modulus

If anyone could help me out here, in particular if you could explain what AB^2 is referring too. Thanks!

2. Let $\displaystyle z=2e^{i\theta_1}$

$\displaystyle w=5e^{i\theta_2}$

we have now
$\displaystyle \frac{w}{z}=2.5e^{i(\theta_2-\theta_1)}$

hence angle between OB and OC is$\displaystyle \theta_1$

which gives$\displaystyle \theta_1=\frac{\pi}{4}$

Now, we have,
$\displaystyle AB^2=39$

I think AB here represents the distance between A and B.
$\displaystyle 5^2+2^2-2.5.2.cos(\theta_2-\theta_1)=39$

$\displaystyle 29-20cos(\theta_2-\frac{\pi}{4})=39$

there is only one unknown, hence it can be solved.

3. Thank you for your help!
But unfortunately your solution is outside the level of my current course (I started complex numbers and my further course last week), I do not recognise what you are doing with e. Do you know any solutions with more basic math?

4. Hi

$\displaystyle arg\left(\frac{w}{z}\right) = arg(w) - arg(z)$

But
$\displaystyle arg\left(\frac{w}{z}\right) = (e_x,OC) [2\pi]$

And
$\displaystyle arg(w) - arg(z) = (e_x,OB) - (e_x,OA) = (OA,OB) [2\pi]$

Therefore
$\displaystyle (e_x,OC) = (OA,OB) [2\pi]$

To find (OA,OB) you can use the formula ABČ = OAČ + OBČ - 2.OA.OB.cos(AOB)

5. Ah I understand a little better now, so the angle AOB is 2pi/3?
How do you suggest finding the lines OA or OB angle with the x axis?
Unfortunately as I said earlier, I don't really understand your lines 2-4.

6. Originally Posted by LHS
Ah I understand a little better now, so the angle AOB is 2pi/3?
Yes
Originally Posted by LHS
How do you suggest finding the lines OA or OB angle with the x axis?
The argument of a complex number z is one measurement of the angle between the x axis (positive values) and the line connecting O to the point represented by the complex number z

This is why arg(w) = (ex,OB)

7. Right, ok, I see, thank for you your help!