Polar Coordinates and Complex Numbers
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Sometimes when multiplying complex numbers, we have to do a lot of computation. Fortunately, when multiplying complex numbers in trigonometric form there is an easy formula we can use to simplify the process. To understand and fully take advantage of multiplying complex numbers, or dividing, we should be able to convert from rectangular to trigonometric form and from trigonometric to rectangular form.
I want to assume that you already know how to multiply complex numbers when they're in rectangular form but how do you multiply them when they're in trigonometric form?
Let's start with 2 general complex numbers z1 and z2. z1 is r times cosine alpha plus i sine alpha and z2 is s times cosine beta plus i sine beta. So r and s are the moduli of the 2 numbers. Remember the absolute value, alpha and beta are the arguments. Let's multiply these 2. z1 times z2 is r times cosine alpha plus i sine alpha times s times cosine beta plus i sine beta. I'm going to collect the 2 moduli out in front. r times s and then these 2 guys I have to multiply together. So I'm going to get cosine alpha cosine beta and notice when I multiply one of the i terms with a non i term, I'm going to have an i in the answer but when I multiply the i terms together, these 2 guys, I'm going to get +i squared sine alpha sine beta. That'll end up being real, so I'll put it in the front. And then we've got i sine alpha cosine beta plus i cosine alpha sine beta. And that's our product. Now this is going to be cosine alpha cosine beta and i squared is -1. So this is cosine alpha cosine beta minus the sine alpha sine beta. Hopefully that looks familiar to you, I'll come back to that in a second. And then let me pull the i out. That sounds painful. Pull the i out and I get sine alpha cosine beta plus cosine alpha sine beta. Sine cosine, cosine sine.
Yes, what are these expressions here. Cosine alpha cosine beta minus sine alpha sine beta is the cosine of alpha plus beta. Right? The cosine of the sum formula. And sine alpha cosine beta plus cosine alpha sine beta, that's the sine of alpha plus beta. Don't forget the i. So i times the sine of alpha plus beta. So this is actually how 2 complex numbers in trig form. You take their moduli and you multiply them. You take their arguments and you add them. It's as simple as that.
So don't forget, to multiply 2 complex numbers, multiply their moduli and add their arguments. This one of the reasons that we like trigonometric form . It's much easier to multiply and it turns out to divide in trig form than in rectangular form.