Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Polar coordinates are a way of finding a point in a plane using a radius and an angle. So R Theta we're going to figure out where the angle is from our pole axis. Our pole axis only goes along the X direction and the positive, and then we have an angle Theta. Where the angle is going counter clockwise R is how far out we've gone on the angle of rotation. These points are not unique, so we can have more than one way of writing a point in polar coordinates. So R, Theta, we're going to rotate around Theta and we're going to come out our distance. This could also be thought of as negative R negative π plus Theta. So if we thought about going to negative π down around here and then we added Theta, we can see that this Theta here would be the same as this Theta in here. So negative π plus Theta negative π Add the Theta in this line here. And if we went negative R, instead of coming out on this line segment, we'd go back through the pole R distance. So these are two points that represent or. These are two ways of writing the same point. So an example, if I wanted 2 2π thirds, I'd go to the 2π thirds and I'd come out to. If I wanted to write every single possibility, we'd have two, 2π thirds plus 2K Pi. That would give me all the adding a full circle or subtracting a full circle where K is an integer. But we could also get points by using this line here. So it would be a -2 going through the pole, negative π + 2π thirds plus all the 2K pi's negative π + 2π thirds is negative π thirds. So if we look here, this is a negative π thirds. And then instead of coming out of +2, which would put us here, we want to go back through the pole. So we're going to go -2. SO 2 2π thirds plus 2K Pi and -2 negative π thirds plus 2K Pi will give us all the location representations for that point equations. If we had R equal A, it's just going to be a circle of radius, the absolute value of a centered at the pole. If we have Theta equaling Theta naughty, it's going to be a line through the pole making an angle of Theta naughty with the polar axis initial ray. So if we had an example of R equal -3, now we're not being told anything about the angle, we're just saying R is -3. So if I put some lines here, if I drew this line here -3 would mean I have to go through the pole backwards by three, IE that blue point right here. If I had the next random angle, instead of +3 coming out this direction, I'm going to go -3 coming out backwards or through the pole, another random angle I just threw up there going -3. When we connect all those rays with the -3 as our radius, we can see that that's a circle. If we had wanted Theta equaling π force, now we don't care what the radius is, so we have an angle of π force. So if radius was one, we'd be here. If radius was 2, we'd be here. Radius 3, radius 4, radius, something in between. So we'd get all these points connected. But if the radius was negative, we'd go through the pole the other direction. So -1 or -2 or -5 halves or etcetera if we want to put two things together. So here we want a radius that's -2 to -1, but the angles that we're looking for are negative Pi 6 through π six. So if we start with thinking we want all the angles from negative Pi 6 through π six, we're talking about all these angles in here. So the next thing we want to realize is we want radiuses that are negative. So if we're talking about all of these in here, now we want the radius to be -2 to -1. So if we're talking -2 to -1, we go through the pole. So here's -1 here's -1 on the circle here and here's -2 So we only want that little piece of shaded region there. We want in between negative PI6 and PI6, but we want a radius that's negative. So we go through the pole, there's my -1, there's my -2. So we want that shaded region right in there. If we wanted to convert cartesian to polar and polar to cartesian, we need to come up with a relationship. So if we thought about putting our polar coordinate on our Cartesian plane, we'd have Theta and R. So our point R Theta would have to equal some point XY in the Cartesian coordinate system. So if we realize that this is a right triangle, here we have cosine Theta being X / r and sine Theta equaling y / r So we could solve each of those for X&Y. So X would be R cosine Theta, Y would be R sine Theta. So we could think the point XY as really being the point R cosine Theta, R sine Theta. Some other relationships we know by Pythagorean theorem X ^2 + y ^2 equal R-squared, and we also know that tangent Theta is just y / X. So now if we start with a polar equation such as R-squared equal 4R cosine Theta, using those four properties, we can figure out what it would look like as a Cartesian equation. R-squared was just our X ^2 + y ^2. Our cosine Theta is just X. So we have X ^2 + y ^2 equal 4X. We know that's a circle. We could subtract the 4X, complete the square, adding four to each side. So we get the quantity X - 2 ^2 + y ^2 equal 4. So the center would be at 20 with a radius of two. If we started with Cartesian and we wanted to go to polar, we could have three X ^2 -, 4 Y squared equal 1, which we know is a hyperbola. So 3 instead of X ^2, we put in what X would be in terms of the polar. So R-squared cosine squared Theta -4 R-squared sine squared Theta equaling 1. Thank you and have a wonderful day.