## TomLikesPhysics 3 years ago I have two complex numbers: z1=a+ib and z2=c+id I am supposed to write them in that form: z=(rcos(theta), rain(theta)). Using that form I need to prove that theta of z3 (z3=z1*z2) is theta of z1 + theta of z2. Using the e^itheta from of complex numbers that would be simple problem but in that component form I can not find a way to prove that.

1. TomLikesPhysics

Here is what I got up till now.

2. Yahoo!

Polar Form .....

3. Yahoo!

Cosx + i sinx = e^(ix) == Euclear Theorem

4. TomLikesPhysics

Aw, I think I am not allowed to use the "e" from. I am supposed to write the complex number as a vector in the form (radius*cos(angle) , radius*sin(angle)). Writing it like that I must prove that the angles add if I multiply two complex numbers.

5. AnimalAin

Let a= r cos theta, b= r sin theta, c= s sin gamma, d= s cos gamma, r= (a^2+b^2)^1/2, s=(c^2+d^2)^1/2 Substitute into your equation, throw a few trig substitutions at it, and declare victory.

6. TomLikesPhysics

Do I am on the right way (on the picture)? I just need to work in the radius?

7. AnimalAin

Put the substitutions above into the equation you have for Z3, and look for a sum of angles trig identity to work backwards to your desired result.

8. TomLikesPhysics

k, I will try to do that. Thank you for your time and help, AnimalAin.

9. AnimalAin

You should end up with $z_3 = rs(\cos (\theta + \gamma)+i \sin(\theta + \gamma)$

10. AnimalAin

Have fun. Do math every day.

11. TomLikesPhysics

I´ll try ;)

12. TomLikesPhysics

Thx, I just did the calculation and it worked out just fine. Thank you for your help, AnimalAin. Now I can claim VICTORY! ;)