 Given the equation of a curve in polar coordinates, we might want to convert it to an equation in rectangular coordinates. To do that, remember the coordinates r, theta give us the rotation angle theta and the distance r from the pole. To get to this point, using horizontal and vertical motions, we need to go x equal to r cosine theta horizontally, then go y equals r sine theta vertically. We can use these equations to transform a polar equation into a rectangular equation. So for example, let's rewrite the equation for the lines y equals 2x and y equals 2x plus 5. So remember to convert from polar to rectangular. We use x equals r cosine theta and y equals r sine theta. And so we have y equals 2x, replacing these will become. And we can simplify this a little. If r is not equal to zero, we can divide by it. And if cosine theta is not equal to zero, we can divide by it. We should worry that dividing by r and by cosine theta gives us an equation different from the original. Since we divided by r, the two equations are different when r is equal to zero. But r equals zero solves r sine theta equals 2r cosine theta. And it also solves tan theta equal to 2, so it's a point on both graphs. And since we divided by cosine theta, the two equations will also be different if cosine theta equals zero. But the point on r sine theta equals 2r cosine theta, where cosine theta is zero, will be. And remember sine and cosine can't simultaneously be zero. So of course sine theta equals zero. Sine theta is not zero, so we can divide by it and find which is on tan theta equals 2. And remember, a difference that makes no difference is no difference. And so tan theta equals 2 is our equation. For y equals 2x plus 5, we'll do those replacements. And we can't really simplify this in any significant way. Or let's consider another pair of equations. So we have x squared plus y squared equals 25, replace, and simplify. And we can simplify a little bit further because we know that cosine squared plus sine squared equals 1. And since r squared equals 25, then we can take the square root of both sides to get r equals 5. You should verify that we don't actually need the negative solution. And similarly, we can substitute. And again, this doesn't simplify very much. Similarly, to get to the point in rectangular coordinates x, y, we want to rotate through an angle of theta equal to arc tangent y divided by x, provided x is not equal to zero. And then go out distance r equals square root x squared plus y squared. Now if x equals zero, we'll either have theta equals pi halves, or theta equals 3 pi halves. But we won't worry about this if we're dealing with a continuous curve. And you might wonder why. And the quick answer is, well, it's calculus. Remember the continuity means that the limit is equal to the function value. So for example, let's try to rewrite in rectangular coordinates r equals 2 minus cosine theta. And so we have substituting. And we can't significantly simplify this equation. So the thing you might notice is the following. As a general rule, a simple equation in rectangular coordinates becomes a complicated equation in polar coordinates. And a simple equation in polar coordinates becomes a complicated equation in rectangular coordinates. So as a general rule, it's not worth converting from one into the other. So why bother? We'll consider that next.