Alright. So in the past couple of videos, we've been spending some time talking about complex numbers in polar form, which looks something like that. Now in this video, we're going to be learning how we can actually do operations on complex numbers, like, say, find the product of 2 complex numbers. Now this might sound like it's going to be really complicated if we have to keep things in polar form, but don't sweat it because it turns out there's actually a very straightforward process for doing this. And once you're familiar with what this equation looks like, I think you're going to find these problems are super straightforward. This is a skill you're going to need to have in this course, so let's just go ahead and jump right into things.

So let's say we have this example, we're asked to find the product of these two complex numbers, and notice that they're both written in polar form. Rather than converting these to rectangular form and then multiplying and converting back, there's actually a much simpler way to do this. All you need to do to multiply two complex numbers in polar form is to multiply the r values. So you start by multiplying the r's out front and then adding the angles. Once you've done this, you just plug everything into this equation, and you'll get your answer. So let's go ahead and try that with the example we have here.

So I'm first going to multiply the 2 r values. I can see that the r value out front here is 3, and the r value out front there is 2. 3 times 2 is 6, so that's what we get. Now we're going to have 6 times the cosine of the 2 angles added together. Now you can see that the first angle is 15 degrees, and see that the second angle is 30 degrees. So we're going to have 15 plus 30 degrees. So all we're doing is adding theta 1 and theta 2, which are the two angles. So we're gonna have the cosine of 15 degrees plus I times the sine of the two angles added together, which is 15 plus 30 degrees.

So we're gonna have this, and then this right here is how we can multiply these two complex numbers in polar form. So let's go ahead and simplify what we have. We have 15 degrees plus 30 degrees, which comes out to 45 degrees. So this is all going to come out to 6 times the cosine of 15 plus 30, which is 45 degrees, plus I times the sine of 45 degrees. And this right here is the product of the 2 complex numbers. So as you can see, once you know this equation, it's a very straightforward process. Just plug in the numbers and simplify.

Now I will say there is something that can get kind of tedious with this, though, and that's writing out the cosines and sines every single time. But it turns out there actually is a shortcut for doing this. So whenever you see this r cos theta + i sin theta, which is the typical thing we see for polar form, you can actually abbreviate this to r cis theta. This is just a more simple and compact way of writing this. So rather than writing our solution as 6 and then with the cosines and sines, we can actually write this whole thing as 6 cis, our angle, which is 45 degrees.

So as you can see, we're able to write this in a much simpler way if we use this kind of cis notation here rather than just writing the cosines and sines every time. So hopefully, this process seems pretty straightforward. But to really make sure we understand this new notation and the process of multiplying complex numbers, let's actually try one more example.

So in this example, we're asked to find the product of the complex numbers below. Now as you can see, we have these two complex numbers, and both of them are in polar form. Now what I'm going to do first is the process that we use for multiplying these two complex numbers up here, so I'm gonna multiply the r values. So we're going to have 4 times 5, which is 20. Now rather than writing out these cosines and sines, I'm actually going to abbreviate all this to -sis. I'm gonna use this new notation we learned about because I think this is gonna be a lot simpler.

So what I'm going to have is cis, and remember we add the two angles. So what we're gonna have is the first angle, which I can see is pi over 6 radians, and I'm going to add this to the second angle, which I can see is pi over 3 radians. So this would be the sum of the 2 angles, and that right there is the product. So all I need to do from here is simplify what we have within this cis notation.

So we're going to have pi over 6 plus pi over 3. Now what I need to do is get like denominators, and I can do that by taking this pi over 3 and multiplying the top and bottom of the fraction by 2. Because what that's gonna give us is pi over 6, which we have on the left side, plus, and we're going to have 2 times pi, which is 2 pi, over 3 times 2, which is 6. Now we have common denominators, so I can just add the numerators.

So we're going to have 20 cis, and then we're going to have the 2 angles added. Well, we have pi plus 2 pi, which is 3 pi, and then that's over 6. Now what I can actually do is simplify this, because 3 over 6 is the same thing as 1 half. This actually reduces. This whole thing is gonna come out to 20cis, and then we're going to have pi over 2, because that's the same thing as 1 half of pi. So we have pi over 2, and that right there is the product of our 2 complex numbers. So as you can see, using this cis notation can make this process a lot simpler, And when it comes to multiplying the numbers, you just need to remember to multiply the r values and add the angles, and that's gonna give you your solution.

So that's how you can solve these types of problems. Thanks for watching. Let's go ahead and move on and get some more practice.