Hey there, everyone, and welcome back. So in the last video, we got introduced to how you could take these complex numbers in this form x+yi, and how you could plot them on these graphs. Now what we're going to be talking about in this video is a new way that you can represent complex numbers using something called polar form. And polar form will take a complex number, and it's a way that you can write it with respect to the distance from the origin, which we call r, and the angle that we make with the real axis, which we call theta. Now this might sound like something that's super scary and complicated, but don't sweat it. Because even though there are a few steps to solving these types of problems, once you know the equations, it's actually pretty intuitive. So let's just go ahead and get right into an example to see how we can deal with this polar form.

So let's say in this example, we're asked to write the complex number 4+3i in polar form, and we can see that we have the point right there. Now if I want to do this, this is the equation you need. X+yi, your complex number, is going to be equal to r times the cosine of theta, and remember r is the distance from the origin, plus i times the sine of theta, where theta right here is the angle that we make with the real axis. So as long as you're able to figure out what r and these theta variables are, you can solve any type of polar form problem. But, of course, the question becomes what exactly are these variables? Well, let's talk about how we can calculate them.

So the way that you can do this is you can recognize that we actually have a familiar shape here that forms. Notice we have a right triangle in this situation, and we've dealt with right triangles many times in this course. Now if I want to calculate this long side of the triangle r, this is equivalent to the hypotenuse, and we know that for finding the hypotenuse we can just use the Pythagorean theorem. C is equal to the square root of a squared plus b squared, where a and b are these two sides. So if I'm dealing with R as the hypotenuse, and we have y and x as these sides, I could also say that R is equal to the square root of x squared plus y squared.

Now likewise, if I want to find this angle, I can again use this right triangle as an example. Recall that the tangent is equal to opposite over adjacent. So going back to this triangle, I can see that the tangent of our angle theta is equal to the opposite side y divided by the adjacent side x. So these are the 2 equations that you need, which will allow you to find r and theta, and then once you find those, you can plug everything into your polar form up here and solve the problem.

r = x 2 + y 2 , θ = tan^{-1}( y x )

So let's go ahead and do this for the example that we have. Well, I'm first gonna use this equation to find r. So we have that r is equal to the square root of x squared plus y squared. I can see that x is going to be the real part, which is 4 squared, plus y, which is the imaginary part, 3 squared. Now the square root of 4 squared plus 3 squared, this turns out to be equal to 5. Now we need to find theta. Well, I can see that theta is going to be yx. So I can see that the tangent of theta is equal to y, the imaginary part 3, divided by x, the real part 4. Now what I can do is use the inverse tangent to find theta because theta is going to be the inverse tangent of the fraction here 3 over 4. That's what happens when we take the inverse tangent on both sides of this equation. If you go ahead and plug this into a calculator, assuming you're in degree mode, you should get approximately 37 degrees as your answer. So that is the angle theta. So we now have our angle theta and we have r, so now I just need to put everything into this polar form equation. So we're going to have r which is 5, we figured that out right there, and that's going to be multiplied by the cosine of our angle, 37 degrees, plus i times the sine of our angle, which is again, 37 degrees. So this right here would be the complex number in polar form. As you can see, we have this form up here, so we did this problem correctly. So that is how you can solve these types of problems we're trying to convert to this new polar form.

Now something else I want to mention about this is that this r that we calculated, this is always straightforward. You just use the Pythagorean theorem. But this theta is not always going to be very straightforward, because the theta that you get, the angle, is going to depend on the quadrant you end up in. Now what do I mean by quadrant? Well, when it comes to your graph, there are 4 quadrants you can end up in. Now in this situation, we ended up in quadrant 1 because you can see that's the only quadrant we had in this example. But if you ever end up in quadrant 2 or 3, you need to add 180 degrees to whatever result that you get. Again, assuming you're in degree mode here. If you ever end up in quadrant 4, you need to add 360 degrees. Now the reason for this is because if you get a point in one of these other quadrants, what's going to happen is you're looking for the total angle. And if we start here on the positive real axis, just finding this angle would not be your total angle. You need to figure out the total angle to get all the way from here to there, or all the way from here to over there. So this is why we need to add a 180 or 360 to get whatever total angle we have, depending on which of these 4 quadrants we end up in. So that is how you can solve these types of problems. Hope you found this video helpful, and let's try getting some more practice with this to really make sure we have it down. Thanks for watching.