 When working with polar coordinates, it's important to remember the trigonometric relationship that connects polar coordinates to Cartesian coordinates, aka rectangular coordinates. It's a trigonometric relationship because they're connected to each other by this typical right triangle diagram that we're going to see right here. Suppose we have a point P in the plane, let's call that point P, and let's call this point right here the origin, so we'll call it the zero vector, like so. What are rectangular coordinates? We'll remind ourselves that if we have a rectangular coordinate system, like x comma y here for P, what we're saying is you go along the horizontal x units and you go along the vertical y units, then you'll find the point P, that's the address you get there. But when it comes to polar coordinates, polar coordinates like r comma theta, well, r is the distance you go from the origin and theta is the angle that gives you that direction right here. So let's convert between polar coordinates and rectangular coordinates. This is the same strategy we used when we talked about vectors given algebraically or vertically, right? If you have some vector v, this is just the magnitude and the direction of the vector, and this would just be the horizontal component and the vertical component of the vector. If we want to think about it in terms of complex numbers, right? We have some complex number here z, then the hypotenuse is just the modulus of the complex number. This is just the argument of the complex number. This of z right there, this would be the real part. This would be the imaginary part. So by this time in trigonometry, this is just the third time we've seen this same conversion of coordinates, complex numbers. You have the Cartesian polar form of the complex numbers. That's actually why we call it the Cartesian and polar form of complex numbers, because it was suggestive of the rectangular and polar coordinates we eventually talked about. We have the geometric and the algebraic forms of vectors. That's just, oh, do we talk about vectors using polar coordinates, the trigonometric form, the geometric form? Or do we talk about vectors using the rectangular form, the Cartesian form, aka the algebraic form? All of those different formulas we saw are really the same formula. And so that's what we're going to use right now as we work with polar equations and Cartesian equations. It's important to remember that the radius here by the Pythagorean equation is r squared equals x squared plus y squared. You can solve for r by taking the square root of both sides, okay? But be aware that r could be positive or negative, so that's why it's better to kind of leave this formula just the Pythagorean relationship. If you need to figure out theta, well, you have tangent of theta is equal to y over x, just by the tangent ratio, like so. If you want to solve for theta, you can take arc tangent of both sides to get theta equals arc tangent of y over x. But I do prefer this formula right here because arc tangent will only give you the reference angle. You have to pay attention to quadrants to get the correct angle right there. So if you know the Cartesian coordinates, then you can switch to polar coordinates using these. But the other way around, x equals r cosine theta and y equals r sine theta. If you know the polar coordinates, you convert to Cartesian formulas using the cosine and sine ratio. So this is just so Catoa trigonometry. So imagine you have a polar equation r squared equals 25. How do you write this in rectangular or Cartesian coordinates? Well, the idea is you have to turn all of the r's into x's and y's. And to do that, you use the Pythagorean relationship. r squared is much better because this would then translate to become x squared plus y squared equals 25. And as we've seen before, this is a circle. This is the circle centered at the origin whose radius is 5. And so the polar equation r squared equals 25 are better yet. This is the same thing as just saying r equals 5. This linear polar equation is actually a circle. And this is sort of like the main thing about polar equations. Polar coordinates are more efficient at circles as opposed to the rectangular coordinate system, which are much better at lines and rectangles. Polar coordinates are actually called circular coordinates because of this observation here. Let's take one that's a little bit more involved. What if you have r squared equals 9 sine 2 theta? Well, the r squared's kind of easy enough. You could write that as x squared and y squared. But what do you do with this 9 sine 2 theta if we want to write this equation using only rectangular coordinates? Well, the conversion formulas we saw on the previous slide rely on theta, tangent theta, sine theta, cosine theta, not this 2 theta business. So in order to make that work, you're going to have to switch it to theta, in which case the double angle identity will be appropriate here. 9 sine 2 theta, this becomes 18 sine theta cosine theta. So that's how you deal with that. So what do you do with sine and cosine? Well, think of your trigonometry, right? Sine is equal to opposite over hypotenuse. So y over r, cosine theta is equal to x over r, adjacent over hypotenuse. So we make those substitutions. You're going to get x squared plus y squared is equal to 18 sine becomes y over r and cosine becomes x over r. Now that's somewhat not the direction we want. Let me amend that. It is the direction we want to go into, but we got rid of theta, but we still have r's. That's what I'm saying. We want just x's and y's. But that's not the end of the world here, 18 yx over r squared. Notice you have 2 r's, r times r's and r squared for which r squared we already know is x squared plus y squared. So you can write this as x squared plus y squared equals 18xy over x squared plus y squared. So this would then be the Cartesian equation we're looking for. Much more complicated than the polar form. You could also clear the denominators if you've done this a little bit differently. You could have had the equation x squared plus y squared, quantity squared is equal to 18xy. That would have also been an acceptable solution here. Now what we've done is we switched from the polar equation to the rectangular equation. Now I don't care if you can graph these things, we'll worry about that another time. Actually the next lecture we'll talk about graphing these things. This might be a little intimidating, but right now we're just practicing the conversion. How do you take a polar equation right in Cartesian coordinates? Basically it's these observations right here. Like so you can get rid of any sines and cosines. You can get rid of any tangents as well, tangent of theta. You get x over y, excuse me, y over x. And since every trigonometric function can be written with just sines and cosines or with a tangent too, you can get rid of all the thetas there. And if you have r's, use the Pythagorean equation and get rid of them. But what if we want to go the other direction? What if we have a Cartesian equation and we want to put it into polar form? Well, the trick here of course is that anytime you see an x, replace it with r cosine theta. And when you see a y, replace it with a r sine theta. So when you see that y on the right hand side, you're just going to replace that with r sine theta. And you can do that on the right hand side as well. You end up with an r squared cosine theta plus r sine squared theta as well. Cosine and sine are squared right there. You can factor out the r squareds, leaving cosine squared plus sine squared. And this is equal to r sine on the right. For which cosine squared plus sine squared is equal to one. This just becomes r squared equals r sine theta, like so. And honestly, I would have jumped here immediately because by the Pythagorean relationship, we know that x squared plus y squared equals r squared. We didn't have to go through this simplifying business because we actually knew that was going to be the case. And so our equations, r squared equals r sine theta. It might be tempting to divide both sides by r and get r equals sine theta. You might be like, oh, that's a simpler equation. For which one has to be cautious in that because you're only allowed to divide both sides by r if r is not equal to zero. So you're kind of assuming that r doesn't equal zero in this situation. For which r could equal zero if sine equals zero, for example. So it turns out the domain of this equation is actually bigger than this one. So really, try not to simplify the equation too much. We just want to convert the equation because, again, the domain for this one would be everything except for r equals zero while the domain of this one was actually good for anything. So be cautious of such a thing there. The proper equation should be r squared equals r sine theta. Let's look at one last example. Let's start or let's end this lecture here with the simplest Cartesian equation which actually forms somewhat of a complicated polar equation. Just take a line x plus y equals four. You might be more used to seeing it in slope intercept form y equals negative x plus four. But nonetheless, this is a linear equation. These are sort of like the simplest equations when it comes to Cartesian coordinates, but what are polar coordinates? Well the strategy we're going to employ is what we did on the previous slide here. X becomes r cosine theta, y becomes r sine theta, and this is equal to four. So in terms of conversion, it's not so horrible. But one thing I want you to kind of keep track of when you're working with these polar equations is why do we like the slope intercept form? What's the big deal, right? Well, it's because really we don't, I mean, while equations are great, we really love functions, functions are sort of king in an algebra setting. And for Cartesian functions, they have the formula y equals f of x, right? The y coordinate is some formula of the x coordinate. So we think of x as the input and y as the output. And so the reason why we're more accustomed to slope intercept form, because this has that format y equals f of x there, y equals negative x plus four. If I know the x coordinate, I compute the y coordinate directly. When it comes to polar equations, polar functions have the form r equals f of theta. And as we start graphing polar equations, we're gonna focus on polar functions next time, where theta is treated as this free variable that can be anything. And then r is then this dependent variable on your choice of theta. So can we solve for r in these equations, all right? That is our sort of ultimate goal when working with a polar equation. Now on the left hand side, there's not much to it, just factor out the r. So you get r times cosine theta plus sine theta, this equals four. And to solve for r, you would get r equals four over cosine theta plus sine theta, like so. And so this would be the polar version, the polar function, which gives us a straight line. Seems very complicated in comparison, right? Cuz you have these trigonometric functions or a fraction of some kind. But this is the thing is polar functions, polar coordinates, are not well equipped to do lines. They can do it, like you see right here, but it's much more cumbersome. Rectangular coordinates are perfect for straight things, like rectangles and lines, flat things, maybe I should call it. Polar coordinates, on the other hand, are better for circular curve things, like circles, right? For which case this function, which gives us a flat line, is much more cumbersome. And this is the reason why we like polar coordinates, is they have advantages that Cartesian coordinates do not. And so while up to this moment, we've probably been used to Cartesian coordinates, that did us great when we were working on flat objects. As we want to think of more circular domains, not intervals, but circular domains, again, this will make much more sense in the next lecture. Polar functions become much more advantageous. And that's why we wanna study them in this lecture here. This actually does bring us to the end of our lecture here. In which case, thanks for watching. If you learned anything about some polar equations, hit the like button. 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