A curve in the plane is given by the parametric equations xsin(t) , Y = cos? (t) , for t in the interval 2 x]In which quadrant of the plane is this curve located?Is...

The given parametric equations define a plane curve. Find an equation in rectangular form that also corresponds to the plane curve. $$x=\sin ^{2} t, y=\cos ^{2} t$$

X in court. You know to sign square tea And then why in Coachella Joe course I scrap t to find the equation Original. Excellent wine. The best way we will be with a line here and we ended up by doing so it again The experts. Why on the left hand side, on the right. And so have that you size squinty plus two course. I square the here we confronted it. You outside and then we have the same square day plus course I square t undersized scrambling courses square. Could you one different? We have a Jew temp's one culture that you and immense and we have experts why you could treat you.

In this question we want to find a rectangular equation generated by this power parametric equation with the running from 0-2 pi. And we want to grab the curve and to indicate the orientation. So first of all we need to find why in terms of X. Now since this is a tree goal parametric question, we would think of squaring the trickle and use the identity Sign Square key plus Coulson Square T. It was the one 2nd. We need to find a domain. Now because he runs from zero to pi. We need to find the valley X. Values for this corresponding T. three. We will graph the cook for we will indicate the orientation. A cup orientation means the direction of the curve. As T increases. Now let's square the X. And the white. And ask them. Now we know that sign square T. But the sine squared is going to be one. So here is the rectangular equation. Now this is a circle. Yeah We center 00 And Radius is one. Mhm. Mhm. Now let's find the domain. Now since our tea is from 0 to 2 pi Sign T will be one full cycle. So is bounded between -1 and one. And our scientists actually our X. So I accessed from -1 to 1. You can see here that we can start drawing it like this. Okay so this is our cool. Now to indicate the orientation let's explore some values of T. As increases So when t equals to zero. We want to find the X. and Y Values 200. So I'll just put that into Collenette from So sub zero into here and here. I will get to a one When T. equals to Highway two. I will get 10 when he goes to pie. Okay 0 -1. When he equals to trip. I want to I will get -10. And when t equals to two pi I will get 01. So it means when to go zero. I start from 01 which is over here and then As T increases to Pi over two, My colleague is 10. So that's over here. That means it moves this way and then from As T. increases to pie is 20-1 and that will be here. So you can see that the orientation is actually what ways? So the orientation is indicated in the re arrow here. Yeah.

All right. Uh We have excess something cause anti Y. Is something scientists. So we want to say something like X squared plus y squared equals one. Except we can't because now that's 16. Co sign square T plus four Sine square T. Which is definitely not one. But what if we did X squared divided by 16. So then that's cosign square T. And then Y squared divided by four. That science square T. Now they do add up to one. So this is actually an ellipse. that makes sense. It's Ko 70 stretched out by a factor for And scientists stretched out by a factor of two. So we have an ellipse with um Semi major axis of length four. Semi minor axis of length to There's 4:00 -400 zero comma two and zero comma negative too. Okay. And then t goes from 0 to 2 pi. It is just the unit circle stretched out. So we are going to trace out the whole thing starting from here um and going all the way around and getting back to here. No initial debatable. So actually I am going to say that that 4:00 is both the initial and terminal points. Um You could debate that there. There aren't any but I think more accurately that one's both

In this question. We want to find a rectangular equation generated by this power parametric equation with the perimeter T. Running from 0 to 2 pi. So firstly we want to find the why in terms of X. Or combination on them. Now, since this is a tree goal parametric question, we will consider squaring the triangle and using the identity sine square T. Last call signs quick. He goes to one 2nd. We'll find a domain the valley values of X. Where T runs from 0 to Pi. That levy will graph the cuff. And hopefully we will indicate the orientation of the curve. The orientation means the direction the curve S. T. Increases. Now let's Now you can see these two and this is treat are not the same. So we actually need to bring them over. So X over two will be signed T. And why over tree this cosign T. Now let's square everybody. And at them. So I have X squared over two square plus Y squared over three square is equal to sine square T. Plus call science square T. Now we know that sign square T. Plus callsign square T. V. One. And our left side, we're just living as this. Now what we have here is an ellipse with the center 00. No let's see the domain for our question. Now we have T from 0-2 pi. That's one full cycle. So our sovereignty Will be bounded between -1 and one. Now let's put a to hear. That means I will multiply a tool To everybody. So my ex is between -2. So I know this is a full lips because of zero to do pine for the tea. So it will look something like this. And It will be something like this where this is to this -2 and this is tree and this is ministry. Yeah. Now to indicate orientation, let's just check out certain values of T. Is usually the increasing value. So when I set T equals to zero, I stopped zero into here to find my X and zero into here to find my wife, I will get 03 X zero wise tree. So zero tree is actually over here. Okay, when I set T equals to power to put it into here to get your ex, my ex will be two and put it into here. My wife will be zero or 20 is actually over here. So you can see that actually the curve direction goes this week, which is clockwise direction. So the blue arrow indicates the orientation of the ground. Okay