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# Parametric equations with the same graph

## Video transcript

in the last video we started with these parametric equations X is equal to 3 cosine of T and Y is equal to 2 sine of T and doing a little bit of algebra we were able to remove the parameter and turn it into an equation that we normally associate with an ellipse we got x squared over 9 plus y squared over 4 is equal to 1 and we graphed it and I'll do that actually I have it from the previous we graphed it and we got something that looked like that right there and then we said well why don't we always just represent it like this well this right here this equation that's just a just in terms of X and y it just represents the shape it doesn't represent the actual path that let's say that that T represents an object's point an object's position at some place in time and we went over that and as time progressed we went around around the ellipse in that case and as time goes forward we kept going around this ellipse but this begs the question is this the only set of parametric equations that when we do a little bit of algebra gives this path can we think of another set that gives that that would also when you do the algebra get here and I think we can so let's try a couple I mean we could do a whole whole bunch I'm actually maybe we will do a whole bunch of them just to see that there at what you'll see is that there's actually an infinite number of parametric equations that have this path so one could be X is equal to 3 cosine of of 2t and the other one is Y is equal to 2 sine of 2t like we did in the last video let's solve for cosine of 2t and solve for sine of 2t so we just have to divide both sides of this is equal both sides of this top equation by 3 you get x over 3 is equal to cosine of 2t divide both sides of this equation by 2 you get Y over 2 is equal to sine of 2t and we'll use the same trig identity as we did before when we did this problem we said that sine squared of t plus cosine squared of T is equal to one well that's true so is this let me see if that's true then we can also say that sine squared of 2t plus cosine squared of 2t is equal to one in fact sine squared of anything here plus cosine squared of anything as long as it's the same anything as we put here I can't have a 2t here and a 3t here but if this is a two and this is a two then that's always going to be equal to one you can just fill in the blank because this is always going to be the same angle so if this is true we can then just take this and substitute it in for cosine of 2t and take the Y over 2 and substitute in for sine of 2t and we get C sine of 2t squared becomes Y over two squared plus cosine of 2t is x over three squared it equals one and that is exactly what we had to begin with that is the same thing as x squared over nine I'm just switching the order here plus y squared over four is equal to one so they both both of these parametric equations both sets of parametric equations have the exact same path so how are they different let's see if we were to take so what in the previous video we we took this yellow one actually I rewrote it here and we did a little table and we plotted the points that it hits as you go around the circle so let's do the same thing for our new set no just since I lost it let me write the two equations here was X is equal to three cosine of 2t and Y is equal to two sine of 2t they make a little table the table there we go and we're going to have a tea I'm going to have an x and a y and let's pick the same values of t zero PI over two and pi when T is zero cosine of two times zero that's cosine of zero which is 1 times 3 it's 3 when T is equal to PI over 2 2 times pi over 2 is PI cosine of PI over 2 is minus 1 minus 1 times 3 is minus 3 when T is equal to PI cosine of 2 times pi is 2 pi cosine of 2 pi same thing as cosine of zero we've so that's 1 1 times 3 is 3 and then on the Y side when T is equal to 0 that's the same thing as sine of 2 times 0 sine of zero 0 times 0 2 is that's 0 PI over 2 sine of 2 times pi over 2 that's the same thing as sine of PI that's also 0 0 times 2 is 0 and then PI sine of 2 pi well that's also 0 so here we get a bunch of zeros so what happened here let me do it in different colors so in our first situation so they both we established that both of both this set of parametric equations and this set of parametric equations they both have the same overall I guess we could say shape of their path they're both this ellipse but we see what we saw in the previous video is when you plot these points at time T equals 0 we will write this this point right here is that right there and then at T is equal to PI over 2 we got right there and then at T is equal to PI we got over here and we did that to establish that the direction of motion if these parametric equations really are describing some type of motion would be counterclockwise now what happens in this scenario so when T is equal to zero we're still at the point three zero so they they at least at T equals zero you can kind of view them as starting at the same point if we assume we're starting at T equals zero you don't have to start at T equals zero I did that a little bit in the last video but you can start at T equals minus a Google or T equals minus affinity so you don't necessarily have a start point but if we assume T is equals zero is the start point you could say that they both start there then at we pick a new color that I haven't used yet then at T is equal to PI over two where are we where minus 3 0 we get all the way oh that's the same color I used before we let me see this let me do this color we're here now notice in the first one when we went from T equals 0 to T equals PI over 2 we went from here to there we went kind of a quarter of the way around the ellipse but now when we went from T equals 0 to PI over 2 where did we go we went halfway around the ellipse we went all the way from there all the way over there and likewise when we went from T equals PI over 2 to T equals PI with this set of parametric equations we went another quarter of the ellipse we went from there to there but here when we go from T equals PI over 2 to T equals PI we go all of this way we go back to the back to the beginning part of our lips so what you see is is that this set of parametric equations has the exact same shape of its path as this set of parametric equations except it's going around it at twice as fast of a rate for every time you know when when when T increases by PI over 2 here we go by we kind of go a quarter way around the ellipse but when T equal increases by PI over 2 here we go half way around the ellipse so the thing to realize and I know I've touched on this before is that even though both of these sets of parametric equations when you do the algebra they can kind of be converted into this shape you lose the formation about where our particle is around as it's rotating around the ellipse or how fast it's rotating around the ellipse and that's why you need these parametric equations we can even set up a parametric equation that goes in the other direction if you instead of having these and you know I encourage you to play with that but if you instead of this if you just put a minus sign right here cosine of minus T and two sine of minus T instead of doing going in that direction it would go in this direction it would go in a clockwise direction so one thing that you've probably been thinking from the beginning is okay Ike was able to go from my parametric equations to this equation of ellipse in terms of just X&Y can you go back the other way could you go from this to this and I think you might realize now that the answer is no because there's no way just with the information that you're given here to know that you should go to this parametric equation or this parametric equation or any of an infinite number of parametric equations I mean anything of the form X is equal to 3 cosine 3 cosine of of really you know anything times T and Y is equal to 3 times cosine of as long as it's the same anything I drew the two swig Lee Marx the same as long as these two things are the same then you will have they'll both converge to this shape so if you have the shape you don't know what parametric equation you can go back to you could make up one but you don't know which one you'll go back to and just to kind of hit the point home of kind of making up a parametric equation sometimes that's asked asked of you so let's just do a very simple example let's start with a normal function of X so let's say that we have the equation y is equal to x squared plus X I just made it up and let's say that you are asked to turn this into a parametric equation and this is often a very hard thing for people because there is not one right answer you can turn it into an arbitrary number of parametric equations I could just make do something really crazy and arbitrary I could say X is when I have Y defined explicitly in terms of X like this I can make X enter anything in terms of T I could say it's cosine of T minus the natural log of T I just made that up that's just a random thing but then if X is this then Y is going to be equal to I just substitute this back in cosine of T minus Ln of t squared plus cosine of t minus Ln of T and we've done we just converted this into a parametric equation or a set of parametric equations I could have also just written x is equal to X is equal to T and then what would y equal Y would be equal to T squared plus plus T now you might say well what's the difference between this parametric equation and this parametric equation well they're both going to have the same shape of their paths it's going to be something like a parabola but the rate and the direction with which they progress on those paths will be very very different and it's actually a very interesting thing to think about there are some paths that you could take where let's say that the you could you could set up a parametric equation let's say that the shape is you know everything we've done so far we've always been going in one direction but you can have a there are scenarios and maybe if I have time I'll make a video and this isn't what you know where let's say the shape of the path is some type of I don't know let's say it's a it's a circle of some kind I'm just going to that's not this so I'm doing a completely different example right now I'm just kind of sometimes you know with the ellipses we had paths that went counterclockwise and then pass that one counter clockwise you can also have paths that kind of oscillate in between move around move back and forth along the thing so there's all sorts of parametric equations that you can define and you can you know you can say if T goes from this to this you're use this set of parametric equations if it's another set of T's using another one so there's all sorts of crazy things you can do to say what happens as you move along the path so the difference not that this is the path of that this is actually more of a parabola but the difference between this and this is how you move along this shape anyway I hope you found that vaguely useful