 So now we get to conversions between these coordinate systems. As a reminder, when we're going 2D positions, we can work with either rectangular coordinates, which is sometimes called Cartesian, and polar. Rectangular is the xy, polar is the rtheta. But there's a connection between the two of these, so let's look at an example. So here I've got a coordinate system drawn out, and I've got a point there. Now I've got my grid for my Cartesian, but I could just as easily put a circle on there. If I've got a circle out at that positions radius, then it's polar coordinates. My rtheta are defined by the length of the line from the center of the circle out to the point, and the angle down here to the x-axis. The xy coordinate systems is how far over and how far up I am, so it's sort of the two sides here of the lines as I draw it out to give me my xy coordinates. I can show those together here, and then what I see is if I shift this x so that rather than drawing it down here, I move it down here to the bottom. I've got my x and y coordinates, my rtheta coordinates, and they're connected through this right triangle here, where the right triangle means that's a 90 degree angle. I've got my angle theta, I've got my hypotenuse, and my two sides of my triangle. So we realize that our rtheta and our xy are connected through a right triangle. So then we should review a little bit of trigonometry for our right triangles. Now one of the standard ways they draw this triangle is with our opposite side, our adjacent side, and our hypotenuse. We sometimes label those A, B, sometimes called C, sometimes called H, and this is our angle theta that we're dealing with down here. Now our standard formulas is that the sine of theta is the opposite over the hypotenuse. The cosine of theta is the adjacent side over the hypotenuse, and then my third trig function, tangent, is the opposite over the adjacent. And some people remember this using Sokotoa, and that's something that some people were taught in geometry. Now we also have Pythagorean theorem, which is if I take the adjacent side squared, the opposite side squared, that's going to give me the hypotenuse squared. You'll see this written a lot of times as A squared plus B squared equals C squared or H squared. Now if I apply this to my triangle here with my x, y, and r theta coordinates, then what I see is that my sine, instead of being the opposite over the hypotenuse, that opposite and the hypotenuse becomes r and y. The adjacent over the hypotenuse for cosine becomes my x over r. And my opposite over my adjacent for the tangent is going to be now y over x for the opposite and the adjacent. And then my adjacent squared plus opposite squared equals hypotenuse squared becomes x squared plus y squared equals r squared. So now we've got our trig functions defined not in terms of the generic ones, but in terms of our actual coordinates for our Cartesian rectangular coordinates and our polar coordinates. Now that we have those basic equations, we can rearrange them to help us find individual things. So if I've got polar coordinates r theta and I want to find my rectangular coordinates x and y, I can rearrange those sine and cosine formulas to tell me that x is going to be r cosine theta and y is going to be r sine theta. Remember, sine dealt with the opposite side, so that associates with y, and cosine dealt with the adjacent side, so that's going to deal with x. If I've got x, y, and I want to find my r and theta, I can rearrange the equations as well. If I take that Pythagorean theorem and rearrange it, I find that it's the square root of x squared plus y squared. To find my theta, I'm going to use the tangent equation, but now I need to do the inverse tangent. Now, this inverse here is sometime called arc tangent. That's not one over tangent, it's the inverse tangent function of y over x. So you can refer back to this video as you're practicing converting between polar and rectangular coordinates. We're going to be using this quite a bit, particularly as we study vectors in physics.