Trigonometry: Multiply Complex Numbers-Trig,Polar

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Multiplying and Dividing Complex Numbers in Trigonometric or Polar Form
So let's see how we can use this polar or trig form for writing complex numbers to actually look at multiplication and division. It turns out that writing the complex numbers in this polar form actually makes multiplication and division a piece of cake. Let me try to show you what we have here.
Let's take two complex numbers but written in this polar or trig form. So that means there's this radius out in front, , times . So what that means is this is the angle that I have to go over on the positive real axis, so I'd go over by , and then this tells me how far to go out. I go out , and that's the complex point . That's what this notation means.
Now suppose I have a second point. I'll call it . And it has as the modulus, and then we have . So I have two different complex numbers, both written in this polar form or in this trig form. And what I want to first of all do is see, what happens if you multiply these things? Well, actually, I will do this out for you, and it's sort of a one-time deal where everyone should see this thing once in their lives but then never again. But let's work it through so we can see what the formula is and see that it makes some sense. So let's right now try to figure out what is equal to? I want to take that thing and multiply it by that thing, so I'm going to actually have to untangle it using the foil method.
Now I'm going to actually distribute the into both of these people and the into both these people just to make my calculations a little bit easier. So I have , and I have to multiply that by . So it's going to be a big old mess basically, but I'll do this so you can watch and just check me. This times this is going to be . Now the inside terms--in fact, let me write the last terms right now. So I'm going to do the first terms, the last terms, and then I'll do the inside and outside terms.
So what's times ? Remember , and . It lifts the radical. So this is going to be a negative. And oh, I have a typo here. This should be an right here. So I have . All of that is just the first terms and the last terms. Now I've got to do all the inside terms.
Now, the inside terms just keep going. Well, the inside terms, I'm going to have an . Here's the . There's no here. Then I have . Then on the outside terms I have just one , so I have a , and then I have again , and then I have . Well, this looks just awful, but it can be simplified quite dramatically if we take it easy and look.
First of all, do you see that there is a factor of here, here, here, and here. Everyone has a factor of , so let me actually pull that right out. times--and now let's see what's left over. Now if I factor out the here, let's see what I get. In fact, I'm just going to write that right on top here so we can look at that together. Let's see what we get. So what I would see is if I factor these out, I'd be left with minus--and then what am I left with? I'm left with this term here after I factor this out, so . Now actually that looks pretty complicated but that actually is the answer to a question. This actually is some cosine of two angles put together somehow. Do you remember the formula? It's actually . Remember, what's the cosine of the sum of two angles? Cosine is greedy so it wants all the cosines first, and then since it's evil, it switches the sign and has all the sines first. So in fact this is actually--this whole complicated thing is just . So in fact that whole complicated thing just boils down to . That's pretty cool. That is really cool.
Now what about this other complicated thing? Well, I've already factored out the so that's taken care of. Notice that these two remaining factors have a common factor of , so I could actually factor out the , and what am I left with? Then I'm left with , so let's write that down. I have . But that has a name. That's actually . Remember how sine works. Sine is very willing to share the limelight, so it's got sine of one angle times cosine of the other, and then it's happy so it's plus, and then it switches the roles. So in fact this is just the sine of the sum of the angles .
So what we just found--this is really cool. If you take these two complex numbers written in polar form or trig form, and you multiply them together--well, you'll never do all this work again, but look at what the answer is. The answer is actually really cool, because the answer can be given in trig form again, and look what you do. To multiply these two numbers, all you do is multiply the 's together, multiply the moduli together--that gives you the new moduli--and what about the angle? What about this term, which is sometimes called the "argument"? What do you do with the argument? You just add the argument, .
So if I want to multiply two complex numbers, it really is easy. All you do is write in the same form, multiply the moduli together and that gives you the modulus of the product, and to get the argument--to get the angle--you just add the angles. That's all multiplication is. So it's really great. You can see the power of polar form or this trig form because multiplying is a breeze. You just multiply these two numbers and add the angles, and that's the answer.
Let's try an example. Suppose I take these two complex numbers: , and I take . Well, then what is ? I'm going to write the answer in polar form, in trig form. I just take this product here, which is 18, times cosine--and then all I do here is add the angles, and so I see and that's the answer. Isn't that easy? And you could actually figure out--this cosine of 120°, you could actually figure that out. You're in the second quadrant. You can actually figure out what the reference angle would be. The reference angle would be 60°, so you can actually figure out cosine and sine. You can actually write this in regular form if you wanted to, in form, or you could just keep the answer like that.
Now, division works the exact same way. The formula is a little different. Let me just tell you how division works so you can long-divide things as well. If you want to long-divide two people that look like this, then what you do is, if you want to take , then the answer--I'm not going to work this out for you but it's the exact same argument that we just looked at--the answer is basically what you would guess. You would take the quotient of these terms, , because before we multiplied, and then we have cosine of--and then what do you think you do? Here we added; now we're going to subtract. So it's going to be . So that's times the sine of that. So that's the formula for dividing. It's the same thing as multiplying except now you divide the terms and subtract the angles. That's all.
So for example, if we come back to this problem, if I want to now ask what is , that's also a breeze because I just take 6 and divide it by 3. . Cosine of--what's the angle? I just subtract here. So if you subtract, what do you see? You see , so it's . And you can keep your form in that form, or if you wanted to you could sort of simplify this a little bit. What's ? Well, remember, you should be thinking about the graphs of these things. , so that's zero. Plus times--and what's ? Well, , so this is actually 2 times zero, which is zero; plus 2 times , so its . So in fact, this complex number divided by that complex number turns out just to be .
So it's really easy to multiply complex numbers and divide complex numbers when they're written in this trigonometric or polar form. All you have to do for multiplication is you multiply the moduli together and then you add the argument, so you add the angles and that gives you the new angle. For division, a similar process. You divide, and then you subtract the angles. Really easy.
Up next well take a look at what happens when you do this a lot, a lot, a lot. I'll see you up next.