INSTRUCTOR: Hi, everyone. Today, we're going to be talking about converting rectangular equations to polar equations and vice versa. So for this first section, we're going to start with rectangular equations. And we're going to convert them to polar equations, which means that all of these x's and y's are going to need to become r's and thetas. And then we're going to solve for r. So our main goal is going to be to solve for r. 

Now, these formulas up here are going to be helpful in getting rid of x and y and getting those r's and thetas to be in the equations. So the first one I have here is x squared plus y squared is equal to r squared. And then I have x equals r cosine theta, and y equals r sine theta. 

So for this first example, I have 2x minus 3y is equal to 5. I'm first going to replace the x with r cosine theta and replace the y with r sine theta. Since I know that my main goal is to solve for r, I'm going to factor in r on this left side here. So if I factor in r out, I'm left with 2 cosine theta minus 3 sine theta. And then I can divide in order to get r by itself. So I end up with r is equal to 5 over 2 cosine theta minus 3 sine theta. 

We'll talk in another video about what the polar equations look like when you actually graph them. For number 2, I have x squared plus y squared is equal to 16. I know that x squared plus y squared is equal to r squared. So I can replace the x squared plus y squared with r squared. And then I square root both sides, and I get r is equal to 4. Spoiler alert, this one is just going to be a graph with a radius of 4. So on my polar axes, if I have 1, 2, 3, 4, the graph r equals 4 is just that circle with a radius of 4 and a center right here. 

For number 3, I have x squared minus 4y squared is equal to 64. I'm going to have to get a tiny bit creative here. I know that x is equal to r cosine theta. So that means x squared is r squared cosine squared. For y squared, I know that that's r squared sine squared. And that's equal to 64. 

If I now factor out an r squared, I'm left with cosine squared minus 4 sine squared I can then divide by the cosine squared theta minus 4 sine squared theta. And then last, I can take the square root. So r is going to be equal to the square root of 64 over cosine squared theta minus 4 sine squared theta. 

Now we're going to look at converting polar equations to rectangular equations. So now all of my r's and thetas are going to have to be in terms of x and y. So for number 1, I'm first going to replace the sine theta with y minus r. So I end up with 4y over r. If I now cross multiply, I have r squared is equal to 4 y. And I know that r squared is equal to x squared plus y squared. So I have x squared plus y squared is equal to 4y. I'm just going to move the 4y over so that I have x squared plus y squared minus 4y equals 0. And there's my rectangular equation for number 1. 

Number 2, r equals 6 over 2 minus sine theta. I'm first going to cross multiply. So I have 2r minus r sine theta is equal to 6. From here, I'm going to replace the sine theta with y over r. These cancel out. So now I have 2r minus y equal to 6. If I add the y over, I have 2r is equal to y plus 6. 

The only thing I need to get rid of now is the r. But the only formula that I have that has an r in it by itself without any sine or cosine is this r squared equals x squared plus y squared. So what I'm going to do is just square both sides of this equation. When I do that, I end up with 4r squared is equal to y squared plus 12y plus 36. 

Now this r squared can become an x squared plus y squared. So I have 4x squared plus 4y squared is equal to y squared plus 12y plus 36. I'm going to get this thing equal to 0. So I'm going to leave the 4x squared. I subtract a y squared. I have 3y squared minus 12y minus 36 equals 0. 

Number 3, tan theta is equal to 2. My formulas up here do not have tan theta. They only have cosine theta and sine theta. But I know that tan theta is equal to sine theta over cosine theta. So I have sine theta over cosine theta is equal to 2. If I cross multiply, I get 2 cosine theta is equal to sine theta. And now I can start using my formulas up here. So cosine theta is x over r. So I have 2x over r. And sine theta is y over r. 

If I cross multiply again, I have 2xr equals y times r, which means my r's just cancel each other out. And I'm left with y is equal to 2x, which in rectangular is just going to give you a nice straight line. That's it for converting polar to rectangular and rectangular to polar equations. If you have any questions, feel free to leave them in the comments below. Have a great day.