 This lesson is on an introduction to polar coordinates, polar equations, and polar graphing. First we'll look at polar coordinates. Well, in order to look at polar coordinates, we have to use polar graph paper. And polar graph paper looks like this. This is one type of polar graph paper, and you'll see it is circular with the rays coming out at the different angles. And of course, it does look like one of the poles. It could also look like this, which is similar, but a little bit different as far as how we represented. But as long as they have the circles with the rays, this would be polar graph paper. So let's go on and look at polar coordinates. Our polar coordinates are written in R and theta. R means how much you go out from the origin, and theta means the angle subtended from there. So instead of x, y, which would be the Cartesian coordinate system, we are R and theta. Eventually we will be going back and forth from x and y to R and theta, but right now we're just going to work on R and theta. So how do we plot these points? Let's say we have the point 2 and pi over 4. That means we go out 2 from the origin to the right along the x-axis, and then subtend an angle of pi over 4. So that would lead us to this point right here. Let's say we want to graph the point negative 3, negative pi over 2. That means to go to the left 3, and you can decide which of these circles you want to be the ones, 2s and 3s, but I just used each circle to be one unit in length. So this is 1, 2, 3 out, and then we want negative pi over 2, so that would bring us back to that point there. So the graphing is R theta. We can also have the same point represented in another way. We can represent this as 2 negative 7 pi over 4. That means to go 2 to the right and back 7 pi over 4. Or we could go 2 to the left and 5 pi over 4, which means 2 to the left and around to that point to 5 pi over 4. Or we could have negative 3 pi over 4 represent that point, which means we go again 2 to the left and back to it at 3 pi over 4. So there are four basic ways to represent this point. Of course, there are many, many more, but we will use this idea when we work on rectangular coordinates being transformed to polar coordinates. And you'll see how this plays out there. But again, that point is represented in many different ways. So you have to stay flexible when you are dealing with polar coordinates. Let's go on. Formulas you need to know. What are they? One is x is equal to R cosine theta. And if you remember, when you represented points in trig, a point p, you represented it as R cosine theta and R sine theta. Again, this comes back to that angle and radius idea. So x can be R cosine theta. That makes y is equal to R sine theta. Both trig ideas. What else do you need to know? Well, the R is equal to plus or minus the square root of x squared plus y squared. Again, a trig idea when you deal with the radius for your point. Another formula you need to know is theta is equal to arc tangent y over x. With these four formulas, you can transform from polar coordinates to Cartesian coordinates and back. So let's try this. The first thing we want to do is convert from the polar coordinate to pi over 4 to its rectangular coordinates in x and y. Well, what formulas do we use? Well, we know that x is equal to R cosine theta. So this becomes 2 cosine pi over 4, which is the square root of 2. Cos cosine of pi over 4 is square root of 2 over 2. Canceling, we get square root of 2. Y is equal to R sine theta, and that goes to sine pi over 4. And that also equals square root of 2. So if we were to graph that point, it would be square root of 2, square root of 2, and lie right there. And of course, 2 pi over 4 we go out to, up to pi over 4, just like we did before. Let's convert the other way. Suppose we had the point negative square root of 3 and 1, and we want to convert it to R theta, the polar coordinates. So R is equal to plus or minus the square root of x squared plus y squared. So that's equal to plus or minus the square root of 3 plus 1, which is equal to 4, and the square root of 4 is 2. So that's plus or minus 2. Theta is equal to the arc tan of 1 over negative square root of 3, and that's equal to negative pi over 6. This coordinate sits in the second quadrant about there. So whatever we put together with our R and theta has to sit in the second quadrant. So there are many possibilities here. We can go negative 2, negative pi over 6, or we can do positive 2 and 5 pi over 6, or any of the other two formulas that will create the point right here. So again, there would be four answers to this. You choose one to be the right answer, or four if you want to do all four. But make sure the point is in the proper place. When you're converting from rectangular to polar. Well, this lends us to what happens with equations. Suppose we are given the equation R is equal to 4 over 1 plus sine theta. If we multiply across, we get R plus R sine theta is equal to 4. R we know to be the square root of x squared plus y squared. R sine theta we know to be y, so that equals 4. If we put the y on the other side, we get the square root of x squared plus y squared is equal to 4 minus y. Square both sides, we get x squared plus y squared is equal to 16 minus 8y plus y squared. And right here the y squareds go out, and we can have y is equal to x squared minus 16 all over negative 8, or 16 minus x squared over 8. Or you could even leave it in the form of 8y is equal to 16 minus x squared whatever form you want to leave it in. But again, we are converting from polar to rectangular. Let's go to the other way to do this, from rectangular to polar. Let's say we have 3x plus 2y is equal to 5, and we want to convert this to polars. x is equal to R cosine theta, y is equal to R sine theta, and that's all equal to 5. If we factor out an R, we get R times 3 cosine theta plus 2 sine theta equal to 5, so R is equal to 5 over 3 cosine theta plus 2 sine theta. These are fairly simple conversions back and forth. They can be far more difficult than this, but I'm just trying to give you a sense of what is going on in these conversions. Next, let's go to graphing. When we graph, we have circles, lamniscates, limousines, cardioids, spirals to work with. So let's look at how we graph using polar curves or polar equations. The first one we're going to look at is the circle. There are three types of circles, really. There's R is equal to some sort of radius or A, and of course that's the circle that lies around the origin. So let's get our calculator out and graph that circle. So first we need to check our mode and go into polar mode. Let's say R is equal to 3. It doesn't look quite circular on the screen, but it is circular on my calculator. So let's look at the window on that, and the window is 0 to 2 pi. So that's theta, and we'll see different pieces of this. First we'll have to figure out what theta is. So it goes from 0 to 2 pi, and that is what creates all the points. And then of course we have the x's and the y's. And of course this is R squared to make circles look like circles. On this we need two things. We not only need the x's and y's for our graphing, we need how theta is handled and how the points on theta are created. And of course it goes from 0 to 2 pi. And when we graphed it, that 0 to 2 pi made one complete revolution, which means that's all we needed to graph it. If I change this to pi and graphed, you see we only get one half of a circle. So with these polar graphs we need to know what creates the entire curve. And in this case of course it is the 2 pi. And there it is again. Our next type of circle is one that says R is equal to sine theta or R is equal to cosine theta. And you will see the sine theta ones are oriented on the y-axis. The cosine is oriented around the x-axis. So let's try that on our calculator. And this time the A is the diameter, not the radius. So let's put R is equal to 3 cosine theta. And graph that. And you see the diameter is 3 units long. And if we changed it to sine, again it is around the y-axis, symmetric to that y-axis with a diameter of 3 units. The next curve is the spiral. And the spiral is either R is equal to A theta or R is equal to A over theta. And let's just try this one on our calculator. R is equal to 2. So either one, R is equal to A theta or R is equal to A over theta. And it's just direction as far as that one is concerned. The next curve we're going to work on are our rose curves. And they are wonderful. They are either R is equal to A sine B theta or R is a cosine B theta. Now if B is an odd number, the number of petals, because it's called a rose curve, there are petals, is the same number as B. And A is the length of each of those petals. If B is even, then we double the number for the petals. So if B is 2, we will have 4 petals. In this particular one, this is the cosine one. Again, it's oriented around the x-axis. This is a sine one. And you notice it's turned because of the way the sine is developed. If the radius is 3 petals, then we have the pole at the symmetry on the y-axis. Let's go through this and just do this on our calculator. Let's say we want 2 cosine 3 theta. We get that rose curve. If we change this to cosine of 4 theta, again, we will double the number. So this time we will get 8 petals. And if you notice, the worm ended when this ended, which means, if I check my window, this ends at 2 pi, which means it takes 2 pi to create the whole curve. And this becomes important when you are doing area under polar curves. Let's go on to another type of curve. And this one is limousine. And there are several different types of limousine. One you see the most is the cardioid or heart-shaped one. And a limousine general equation is r is equal to a plus or minus b sine theta. Or r is equal to a plus or minus b cosine theta. Now, how do these work? Well, the cardioid has a and b equal to each other. The in a loop type has a less than b. The one that has the loop extending out is a is between b and 2b. a is greater than b, but less than 2b. And one that looks like an oval is a is greater than equal to 2b. The two that you see the most are the cardioid and the limousine with the inner loop. Let's do a cardioid on our calculator. 2 plus 2 cosine. And again, this one's oriented around the x-axis. So when it comes out, we will see it's the cardioid oriented around the x-axis. Now, there are important things on this. It extends out four units. That's the 2 plus 2. It goes up two units and down two units on the y-axis. That is just the 2 because when cosine is pi over 2, that part of the equation becomes zero. And then it comes around to the point where it is zero. So again, the graphing on this. Let's just redo that graph and look at the worm at the same time. And this too ended at 2pi. So we can see the window goes from zero to 2pi. The other one, the inner loop one where we have A is less than B. Let's put a 1 in here and graph that. Again, it's going to extend out three units this time. And then the inner loop is the 2 minus the 1, which is 1. So that is one unit long. And again, it extends out three units, the 2 plus the 1. This is a general form for that. Let's go on to another type of curve. And this one is your lamniscate. And that one says R squared is equal to A squared sine 2 theta or R squared is equal to A squared cosine 2 theta. The cosine one again is oriented around the, or symmetric to the x-axis. The sine one just turns 45 degrees. So let's try one of those. Now remember, this one needs a square root. So we need, let's clear that out, second square root. And pick 4 sine 2 theta and graph that. And you'll see it is turned 45 degrees. And if you were to plot points in any of these, you can see where all the graphs come from just by seeing their plotting points. But I believe you'd need a general idea of what's going on with these graphs. And if you have time, plot those points and see how they are graphed out through those plotted points. This gives you a quick introduction to polar curves, polar equations, polar coordinates, polar graphing. Hope this helps as you do your calculus work in this particular unit. This concludes your lesson on introduction to polix.