>> In this section we'll be talking about a technique of identifying points on a coordinate plane, not so much according to horizontal and vertical distances as we've been accustomed to but in terms of an angle and a distance along the terminal side of that angle. Here's what I'm talking about. Suppose we have a point 3, 4 on the coordinate plane, and suppose we think about generating an angle, and from standard position to that point, that is an angle that we might call theta like this. And then along this ray, we define a distance to that point. You see it's just another way to describe the same position in the coordinate plane. Now we're gonna call the distance here R and the angle will be theta, and we can understand very quickly how this angle and theta, excuse me, this angle and distance R can be calculated. The coordinates here are 3, 4, so that's 3 and 4 on a little right triangle. And let's see the R value would be calculated according to the Pythagorean relationship R squared is equal to 3 squared plus 4 squared or R is the square root of 3 squared plus 4 squared. Let's put that down over here. R squared is X squared plus Y squared. R then is the square root of the sum of those squares. And for this particular problem, R is 3 squared plus 4 squared and then the radical well that's 9 and 16 that's 25, so the distance R is 5 for this problem. Now it's important for us to know the general idea for the calculation of R, though. Now for theta, it's a similar kind of thing. We can just use past experience to understand how to find the theta and we have information about opposite side and adjacent side next tangent ratio so that tangent of theta is 4 over 3, and more generally it's Y over X in this kind of configuration. So tangent theta is Y over X and this is how we make the calculation for theta. In our problem it's 4 over 3. Inverse tangents then of 4/3 would give us a theta of approximately 53 degrees. Now this same point then can be described in terms of an ordered pair but not horizontal and vertical distance. It's the ordered pair of a distance of 5 and a theta of 53 degrees. A lot of times in using this polar form, this is a called polar form or maybe trigonometric form. In using this polar form, it's handy to use a polar coordinate plane like this. Now notice it's not so much an X, Y plane but it's a plane on which we can see angles very easily and measure distances along those angles. It really doesn't make any difference if you have polar coordinate graph paper, but if you have access to that, you might pick some up because it's really neat to find points on this polar coordinate plane. Let's do that with a few points here. If we're talking about 3 and Pi over 4, then we would first look at an angle of Pi over 4 and we know that to be a 45-degree angle. And that would be this angle and you see we would have this ray, and then along that ray we're gonna go a distance of 3. Well, let's see. Now I have marked off, I've made marks here along the horizontal axis and I'm just kinda follow those up on a ray. So it's a distance of 1, 2, 3, so here's the 1, 2, 3. So here then is the point 3, Pi over 4.