Welcome back, everyone. So recall in the last video how we learned about how we can take products of complex numbers in polar form, and the way we do this is by multiplying the r values and then adding the angles together, and then putting them all together in this equation. Well, in this video, we're now going to learn how we can do what is in essence the opposite of multiplying 2 complex numbers in polar form, which is going to be finding the quotient or dividing complex numbers. Now something you might be happy to hear about is even though this process may sound scary, it's actually very straightforward if you know how to multiply complex numbers. And this is a skill you're going to need to have in this course, so let's just jump right into an example to see how we can do this. So let's say we have these two complex numbers right here, and we're asked to find their quotient, which is the same thing as dividing them. Now, the nice thing about dividing complex numbers in polar form is that it's literally the same idea as multiplication, but with opposite operations. So let's think about this for a second. If I have to divide these complex numbers, well, what do we do when multiplying them? Well, multiplying complex numbers, we multiplied the r values. So, the opposite of multiplication, the way I like to think of it, it's division. So rather than multiplying the r values, we would divide the r values. So r1 divided by r2. And recall that for the multiplication or the product, we add the angles. So when it comes to dividing, we subtract the angles. So literally, it's just opposite operations. You divide the r values and subtract the angles rather than multiplication and addition. So it's straightforward. So let's go ahead and do it. So what I'm first going to do is divide the r values. So we're gonna have 6 divided by 3, that's the 2 r values, which is 2. And then that's going to be multiplied by we have the cosines and sines, but recall that we can actually write this in a more simple way. We can write this as cis. Right? That's a more compact way to write this that we learned in the previous video. So, we're going to have cis, the two angles subtracted. Well, I can see that the first angle is 45 degrees and that's going to be minus the second angle which I can see is 15 degrees. So we're going to have 45 minus 15 which is 30. So we're going to have 2 cis 30 degrees. And this right here is the solution to our problem. So that's how you can divide 2 complex numbers or find their quotient. Again, it's the same equation as multiplication, just with opposite operations. Now let's go ahead and try another example to really make sure that we have this down. So what I'm going to do is use this situation where we're told these are the two complex numbers that we have z1 and z2, and we're asked to find their quotient. So the first thing I'm going to do is I'm going to divide the r values out in front. So to find z1 divided by z2, we're going to first divide r1 and r2. r1 I can see is 5, and r2 is 4. So that's going to be what we have out in front, and then we're going to have this cis because that's the same thing as the cosines and sines, and we're going to have the 2 angles subtracted. So you can see the first angle is pi over 3. So we're going to have pi over 3, and then we're subtracting the angles, and that's going to be minus pi over 9 radians. And that right there would be what we have. So let's just go ahead and simplify this. So we're going to have 5 fourths cis, and we're going to have pi over 3 minus pi over 9. Now what I want to do is get like denominators. I can do that by taking the first fraction that I see and multiplying the top and bottom by 3. What that's going to give us is 3 times pi, which is 3 pi, divided by 3 times 3, which is 9. Then we won't have minus pi over 9 that stays the same. So this is what we have in the parentheses. Now from here, we're going to have 5 fourths cis, and then we're going to have 3 pi minus pi which is 2 pi divided by the common denominator of 9. So this is the solution right here, and that's how you can solve these types of problems. So, hope you found this video helpful. Thanks for watching, and let's move on and get some more practice.