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

## 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.

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.

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:

$$|x|^2 = x\overline{x}$$
$$|y|^2 = y\overline{y}$$

$$|xy|^2 = xy\overline{xy}=x\overline{x}y\overline{y}=|x|^2|y|^2$$

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