X=r(cos u+i sin u) and y=t(cos v + i sin v)

  • Thread starter raiders06
  • Start date
2
0

Main Question or Discussion Point

I need help getting this one started... PLEASE...

Given x=r(cos U + i sin u) and y =t(cos v + i sin v):
Prove tha tthe modulus of (xy) is the product of their moduli and that the amplitude of (xy) is the sum of their amplitudes.
 

Answers and Replies

207
0
I dont really know the answer to this question out-right, though it seems like that they are taking x,y in the complex field to a polar coordinate mapping. When you multiply two elements in this way the length's are a product and the angles add together which can be verified by straight multiplication of the variables xy.

hince: xy = rt(cos(u+v) + isin(u+v))

Graphically this looks like x/y lengths multiplied with their angles added together.

That should give you a good insight into what the modulus/length and the amplitude/direction should be.

Edit:

Also: when proving the xy = "" portion I posted, it may be helpful to look up trig identities for cos(x+y) and sin(x+y) as it will allow you to make needed substitutions :P.
 
101
1
Maybe use Euler's formlua?
 
The modulus of a complex number is given by multiplying the number by its complex conjugate and taking the square root. So:

[tex]|x|^2 = x\overline{x}[/tex]
[tex]|y|^2 = y\overline{y}[/tex]

[tex]|xy|^2 = xy\overline{xy}=x\overline{x}y\overline{y}=|x|^2|y|^2[/tex]

This works because complex multiplication is commutative(order doesn't matter). Now just take the square root of both sides and you're done.
 

Related Threads for: X=r(cos u+i sin u) and y=t(cos v + i sin v)

Replies
1
Views
4K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
4
Views
48K
  • Last Post
Replies
9
Views
886
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
12
Views
2K
Top