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Dividing Complex Numbers - Concept
Sometimes when dividing complex numbers, we have to do a lot of computation. Fortunately, when dividing complex numbers in trigonometric form there is an easy formula we can use to simplify the process. To understand and fully take advantage of dividing complex numbers, or multiplying, we should be able to convert from rectangular to trigonometric form and from trigonometric to rectangular form.
I'm going to assume you already know how to divide complex numbers when they're in rectangular form but how do you divide complex numbers when they are in trig form?
Here are 2 general complex numbers, z1=r times cosine alpha plus i sine alpha and z2=s times cosine beta plus i sine beta. I'm going to divide these 2 and come up with a formula. So let me start with z1 over z2. That's going to be r cosine alpha plus i sine alpha s cosine beta plus i sine beta. And you remember the trick when you're dividing complex numbers to multiply the top and bottom by the conjugate of the denominator. So just this part, you don't have to worry about the s.
The conjugate of this denominator is cosine beta minus i sine beta. I have to multiply the top and bottom. And this product, because it will end up giving me a difference of squares will actually end up being a real number. Let's see that. So we have s. Actually let me pull the r over s out in front. So we got the r over s here times and then cosine beta plus i sine beta times cosine beta minus i sine beta. Is cosine squared beta minus i squared sine squared beta. And then in the top, we've got cosine alpha cosine beta and the other real term is going to be the product of these 2 guys. Minus i squared sine alpha sine beta. And then i sine alpha cosine beta and minus i cosine alpha sine beta. Okay, that's a lot. So, let's take a look at this. This denominator, remember that i squared is -1 so -i squared is +1, this is cosine squared beta plus sine squared beta by the Pythagorean identity, this denominator is 1 and so we're left with just the numerator, r over s times that numerator. So let me copy that up here. z1 over z2 is r over s times and again i squared is -1 so this will become plus sine alpha sine beta. Cosine alpha cosine beta. Cosine alpha cosine beta plus sine alpha sine beta, and then actually I'm going to need a bracket here, plus i times, I'm going to pull the i out of these 2 terms and I'll be left with sine alpha cosine beta minus cosine alpha sine beta. Sine alpha cosine beta minus cosine alpha sine beta. Okay, this is it.
Finally, what's cosine alpha cosine beta plus sine alpha sine beta? This is one of the addition formulas for cosine, right? It's the cosine of a difference. Remember the cosine of a difference has a plus sign in it. So it's cosine of alpha minus beta and then plus i times and this is the sine of alpha minus beta. And what this reveals is, if you want to divide 2 numbers, 2 complex numbers you divide their moduli and you subtract their arguments. That's the thing to remember, to divide 2 complex numbers, divide the moduli and subtract their arguments.