Fiendish algebra problem

I have a complex variable X of the form X=A*cos(theta)+i*A*sin(theta)
where A and theta are the amplitude and phase of X respectively.

I have an equation relating X to a real variable Y:

X=(a-b+i*a*b*Y)/(a+b+i*a*b*Y) (equation 1)

where a and b are real constants.

If I plot out the locus of X as Y varies, I can use geometry to prove the following relationship between the amplitude (A) and phase (theta) of X:

A=(1/(1+a/b))*cos(theta) +or- ((1/(1+a/b))*(cos(theta))^2-((1-a/b)/(1+a/b)))^0.5

I desperately need to prove this algebraically from equation 1, but am a lowly engineer and so am unable to. Any help achieving this would be greatly apreciated.

[wrinkles nose] Sniff sniff! Is that homework I smell?



There is a standard way to solving these equations, employed whenever you have i in the denominator. I'll tell you what it is, then you can do it :).

In your equation for X, multiply the RHS by 1. RHS x 1 = RHS, so this is OK.
But we will write 1 as (a+b-iabY)/(a+b-iabY). Clearly this is 1, but when you do the multiplication and expand, all the i terms in the denominator will disappear. As if by magic. By swapping the sign and multiplying, you ensure that this is so.

So, expand X = (a-b+iabY)(a+b-iabY) / (a+b+iabY)(a+b-iabY)

Then you can collect the terms on top of the equation into those with an i and those without.
You now have something of the form X = G + iH

and since you know that X = A cos theta + i A sin theta, you now know that G = A cos theta and H = A sin theta.

And then you rearrange, and that's it!


Have fun!

zaza

Thank you for the reply. Sorry I should have been more clear in my question. I am aware that the rout to take is to multiply by the complex conjugate of the denominator and equate real and imaginary parts.

The problem is this: you will notice that the equation I am seeking to derive relating amplitude to phase is independent of the variable Y. When I multiply by complex conjugate and equate real and imaginary parts, I can get an equation relating amplitude to phase, but it is proving very difficult to remove the variable Y from it.

I hope that I have made some sort of sense there.