Question

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.

Answer 1

Here is what I got up till now.

Answer 2

Polar Form .....

Answer 3

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

Answer 4

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.

Answer 5

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.

Answer 6

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

Answer 7

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.

Answer 8

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

Answer 9

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

Answer 10

Have fun. Do math every day.

Answer 11

I´ll try ;)

Answer 12

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