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TomLikesPhysics
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.
Here is what I got up till now.
Cosx + i sinx = e^(ix) == Euclear Theorem
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.
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.
Do I am on the right way (on the picture)? I just need to work in the radius?
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.
k, I will try to do that. Thank you for your time and help, AnimalAin.
You should end up with \[z_3 = rs(\cos (\theta + \gamma)+i \sin(\theta + \gamma)\]
Have fun. Do math every day.
Thx, I just did the calculation and it worked out just fine. Thank you for your help, AnimalAin. Now I can claim VICTORY! ;)