Welcome back, everyone. So in the previous video, we learned about how you could write complex numbers in polar form, which looks like this. Now, in this video, we're actually going to see how you can take complex numbers in this polar form and convert them back to their original form \(x + yi\). And this form, \(x + yi\), we actually call rectangular form. If this sounds complicated or abstract, don't sweat it. Because I think you're going to find that converting from polar to rectangular form is actually very straightforward. It's even more straightforward going in this direction than converting to polar form in the first place, in my opinion. And I think you might find that too. So without further ado, let's just go ahead and jump right into things.

Now let's say you have this complex number, and it's clear that we have polar form here since we see that we have an \(r\) value and we have an angle. If we want to take this and convert it back to rectangular form or just convert it there in the first place, all you really need to do is take whatever your \(r\) value is, and you need to distribute it. And that's really the main strategy for converting back to rectangular form. You distribute the \(r\)'s and then you evaluate whatever numbers you have, and that will tell you what your rectangular form is. So let's go ahead and try it right here.

We're given the example where we're asked to convert the complex number from forward to rectangular form and to identify the \(x\) and \(y\) values, or basically the real part and the imaginary part. So what I'm going to do is take this 5 and just distribute it into these parentheses. So if I do this, we're going to have \(5 \times \cos(37\degree)\) plus \(i \times 5 \times \sin(37\degree)\), and there's our distribution step right there. Now, \(5 \times \cos(37\degree)\) you can actually put that into a calculator. And if you do this, this should approximately come out to be 4. Now if you take \(5 \times \sin(37\degree)\) and you put that into a calculator you should get about 3. So \(4 + 3i\) is the complex number we get. And as we can see, the real part is 4 and the imaginary part is 3. So this here is a complex number and these are the solutions for \(x\) and \(y\). So as you can see, it's a very straightforward, very quick process to do this conversion.

Now to make sure that we know how to do this well, let's try one more example where this time we're dealing with radians. So as to convert the complex number from polar to rectangular form and as you can see here, we have our angle in radians this time. But, thankfully, we can use the same strategy where we just distribute the \(r\) value. So I'm first going to start by taking this 8, and I'm going to distribute it into these parentheses. So we're going to have \(8 \times \cos\left(\frac{\pi}{6}\right)\) minus \(i \times 8 \times \sin\left(\frac{\pi}{6}\right)\). So that's our first step. Now, from here, you have a couple of options. You could take the \(8 \times \cos\left(\frac{\pi}{6}\right)\) and \(8 \times \sin\left(\frac{\pi}{6}\right)\), and you could plug that into your calculator, but there's something we should recognize here. Both the cosine and the sine of \(\frac{\pi}{6}\) are actually values we can find on the unit circle. For the cosine, the cosine of \(\frac{\pi}{6}\) is the square root of 3 over 2. And the sine of \(\frac{\pi}{6}\) is actually 1 half. So if I rewrite this complex number which I'll do over here, we're going to have that \(z\) is equal to \(8\) multiplied by this value, \(\sqrt{3}/2\), And then this is going to be minus, and then we're going to have \(i\), and then that's going to be times \(8\), that's going to be multiplied by this quantity which is \(1/2\). So all I need to do is evaluate the numbers that I see. As you can see, I didn't even need to use my calculator for this one. So we're going to have that \(z\) is equal to \(8 \times \sqrt{3}/2\). Now, \(8\) divided by \(2\) is \(4\), so we're going to have \(4\sqrt{3}\), that's gonna be minus, then we'll have \(i\), then \(8\) divided by \(2\) is \(4\), so that's gonna be \(i \times 4\). So what our complex number becomes is \(4\sqrt{3} - 4i\). And this right here is our solution for \(z\) and that is the complex number in rectangular form. So as you can see, it's a very straightforward process. Just distribute and figure out what the sines and cosines are. You can do this on a calculator or with a unit circle, and then you can get your final answer. So that is how you can solve these types of problems. Hope you found this video helpful. Thanks for watching.