Everyone, in this example, we've got 2 parametric equations. We've got x(t)=3∙cos(t) and y=2∙sin(t). Now remember, these involve trigonometric functions, so it's always best to rather than first graph it, we're actually just going to first eliminate the parameter so we can get the rectangular equation, and then we can graph the plane curve after we've gotten rid of that parameter t. Alright? Let's go ahead and get started here. Remember, these types of problems when you're eliminating the parameter always boil down to trying to get something in terms of the Pythagorean identity cos2(t) + sin2(t) = 1 . I've got equations here that involve cosine and sine, so I'm just going to have to rewrite those things in terms of the x's and y's. Alright?

So how do we do this? Well, I've got that x=3∙cos(t). I want cos2(t). Let's bring the 3 over to the other side. I'm going to divide by 3, and then I'm basically just going to go ahead and take the square roots. Or sorry. I'm going to square both sides. So in other words, I'm going to get x32=cos2. Alright? So now what happens is every time I see, or wherever I see a cos²(t), I can actually just replace it with x32. Alright? Same thing over here with the y axis or the sine of t. Alright? So that y=2∙sin(t). If I want to rearrange, I've got to divide 2 on both sides, and then I'll just square both sides. So I get y22=sin2t. Alright? So now what happens is wherever I see sin²(t), I can actually just replace it with this expression over here. So if I combine all of that stuff, what I end up with is that x32 + y22 =1. So I've eliminated the parameter. This is going to be my rectangular equation over here. And so what happens is this is my answer. And now that I've eliminated the parameter, now I can go ahead and graph this plain curve because I already know what this thing is going to look like, or I have an idea of what this thing is going to look like because now I have the rectangular equation. If this doesn't look familiar to you, we have x2 + y2, but you have these little numbers here that are inside of the denominators. This is actually the equation of an ellipse. We've seen this before when we talk about conic sections. This is an ellipse because you have these two numbers over here that are not the same that are underneath the x's and y's, and this is going to be an ellipse. It's centered at 0. So it's centered at 0, and basically, the 3 and the 2 are going to make up the semi-major and semi-minor axes or the major-minor axes. So, in other words, because the x is bigger, this is going to be a sort of an oblong oval that's going to be wider in the x than it is in the y. And it's basically to go 3 in the x and then 2 in the y like this. So, basically, what our graph is going to look like is it's going to go from 3 to -3 on the x-axis and then 2 to -2 on the y-axis. And If you connect these things, what you end up getting is you'll end up getting something that looks like this. Alright. So are we actually done here? Well, in most cases, we would be because usually what's going to happen inside of these problems with trigonometric functions is that your parameter t is going to go from 0 to 2π, in which case you're just going to go all the way around the circle. We're not actually quite done here yet because we have to take a look at the restriction of t. t only goes from π all the way to 2π. It doesn't go from 0 all the way to 2π. So what happens here is that this piece, the top half of the ellipse, this would be from 0 to 2π. Oh, sorry. This would be 0 ≤ t ≤ π. And then what happens is the bottom half of the ellipse would be π < t < 2π. Think of just the way that the unit circle works. Right? Once you go all the way to π, that's going to be all the way on the negative x-axis. And then all the way around again would be going back to 2π. So what happens here in this problem is we actually can't graph the top half of the ellipse because that's not inside of the parameter. So we don't have the entire interval from 0 to 2π. We only have from π to 2π. So the only part that you can actually graph here is going to be the bottom half of the ellipse. Alright? That's always an important thing to consider when you are doing these types of problems is that sometimes the t restriction won't actually get you the entire circle or the entire ellipse or whatever shape it is. Alright, folks. So that's it for this one. Let me know if you have any questions.