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Removing the parameter in parametric equations (example 2)

Video transcript

let's see if we can remove the parameter T from a slightly more interesting example so let's say that X is equal to three times the cosine of T and Y is equal to two times the sine of T we can try to remove the parameter the same way we did in the previous video where we can solve for T in terms of either X or Y and then substitute back in and we'll do that I'll do that but I want to do that first just to show you that it kind of leads to a hairy or an unintuitive answer so if we solve for let's solve for T here and we can do it either one they're equally complex so if we will solve for T here we would say divide both sides by two you would get Y over two is equal to sine of T and then you would take the arc sine of both sides or the inverse sine of both sides and you would get I like writing arc sine because inverse sine off off people often confuse it with an exponent taking it to the negative one power so arc sine of Y over two is equal to T now actually let me do that little aside there I should probably do it at the trigonometry playlist but it's it's a good thing to hit home because I think people get confused sine so arc sine arc sine of anything of let's say Y this is the other way of writing that is sine minus 1 of Y these two things are equivalent when they're normally used but I don't like using this notation most of the time because it can be ambiguous this could mean sine of Y to the negative one power this could mean sine of Y to the negative one power which equals one over sine of Y and arc sine and this are definitely not the same thing so you want to be very careful there to make sure that you don't get confused when someone writes an inverse sine like this that they're not actually taking sine of Y to the negative one power on the other hand if someone were to write sine squared of Y this is unambiguously the same thing as sine of Y squared in fact I wish this was the more conventional notation because it wouldn't make people think of two and minus one there and and of course that's just sine of Y squared so it can be very ambiguous and of course if this was a negative and this would be a minus two and then this really would be one over sine of Y squared so I just that's why just a long-winded way of explaining why I wrote arc sine instead of sine inverse sine right there needless to say let's get back to the problem so we've solved for T in terms of Y now we can substitute back here and we get an expression for X in terms of Y so we get X is equal to three times the cosine of T we just solve for T T is this thing right here so it's the cosine of arc sine arc sine of Y over 2 and we have eliminated the parameter but this is a very non-intuitive equation we could have done the other way we could have solved for Y in terms of X and we've gotten you know the sine of the arc cosine it would have been equally hairy or not intuitive but in either way we did remove the parameter so I guess we could mildly pat ourselves on the back but that's not the purpose of this video the purpose of this video is to see if there's any way we can remove the parameter that leads to a more intuitive equation involving X and y and what we're going to do is I guess you can call it a bit of a trick but it's something that shows up a lot especially when you deal with polar coordinates and you might want to watch my polar coordinate videos because this is essentially touches on that but if I said let me rewrite them X is equal to 3 cosine of T and Y is equal to 2 sine of T so what we can do is just you know think about well how can we write this and you know close sine of T and sine of T how can we relate them and the first thing that comes to my mind is just the unit circle or or to some degree the most basic of all of the trigonometric identities and that is that the cosine squared of T plus the sine squared of T is equal to 1 this comes from the unit circle I explained it in the it's circle video and that's because the equation for the unit circle is x squared plus y squared is equal to one the cosine of the angle is the x-coordinate the sine of the angle is the y-coordinate so on and so forth but this is our trig identity you don't have to think about it too much right now just I guess know that it's true and watch some of the other videos if you want to proven that it's true but if we can somehow replace this cosine squared with some expression in X and replace the signs sine or the sine squared with some expression Y we'd be done right and then we would have it equaling one and that shouldn't be too hard we can rewrite this we can set cosine of t equal to something in X and we can set sine of T equal in something and Y and so let's do that so we divide both sides of this equation by 3 you get x over 3 is equal to cosine of t and if you divide both sides of this equation by 2 you get Y over 2 is equal to sine of T and then we can use this trigonometric equate identity we can just instead of the cosine of T we can substitute x over 3 instead of the sine of T we can substitute Y over 2 and you get you get x over 3 squared that's that right there that's just cosine of T squared plus y over 2 squared that's just sine of T squared is equal to 1 and now this is starting to look a lot better than this this I have no idea what this is but hopefully if you've watched the conic section videos you can already recognize that this is starting to look like an ellipse we could simplify it a little bit we could say this is equal to x squared over 9 plus y squared over 4 is equal to 1 and if we were to graph this ellipse and we will actually graph it we get let me draw my axes I'm using this blue color a little bit too much getting it's getting monotonous ok let me use a purple so that's our x-axis that's our y-axis the major axis is in the X direction because the denominator here is larger than that one and it's the semi the the semi major radius is going to be the square root of this so it's 3 1 2 3 in that direction 1 2 3 I know I'm centered at 0 because neither of these are shifted you should watch the conic section videos if this sounds unfamiliar to you and the semi-minor radius is the square root of 4 so that's 2 so we get 1/2 and 1/2 let me see if I can draw this ellipse so it would look something like something like that there you go so just like that by eliminating the parameter T we got this equation in a form that we immediately were able to recognize as a lift when I just look at that unless you deal with parametric equations or maybe polar coordinates a lot it's not obvious that this is the parametric equation for an ellipse but this once you learn about conic sections it's pretty clear it's an ellipse and it's easy to draw that ellipse but in in in removing the T and from going from these equations up here and from going from that to that like in the last video we lost information we lost 1 what is the direction that we move in as T increases and we also don't know what point on this ellipse we are at any given time T so to do that let's make our little table let's make our little table so let's take some values of T so we'll make a little table T X and Y and it's good to pick values of T remember let me rewrite the equations again so we didn't lose it X was equal to three cosine of T and Y it was equal to two sine of T it's going to take values of T that it's easy where it's easy to figure out what the cosine and sine are and without using a calculator we're assuming the T is in radians just for simplicity so let's pick T is equal to 0 T is equal to PI over 2 that's 90 degrees and degrees and T is equal to PI and so what is X when T is equal to zero well cosine of zero is one times three that's three what's X when T is you have PI over two cosine of PI over 2 is 0 0 times 3 is 0 and what's X equal when T is equal to PI cosine of PI is minus one minus one times three is minus three fair enough now let's do the Y's when T is 0 what is y sine of zero is zero so 2 times zero is zero when T is PI over 2 sine of PI over 2 is 1 1 times 2 is 2 and when T is PI sine of PI that's sine of 180 degrees that's zeros two times zero is zero so let's plot these points when time is zero we're at the point 3 comma 0 so 3 comma 0 3 comma 0 is right there this is T equals 0 when T increases by PI over 2 or if this was seconds PI over 2 seconds it's like 1.7 something seconds so at T equals PI over 2 where the point zero comma two we're at zero comma 2 we're right over here so this is at T is equal to PI over 2 and then when T increases a little bit more when T's increased a little bit more when we're at T is equal to PI we're at the point minus three zero so minus 3 0 we're here so this is T is equal to PI or you know we could write 3.14159 seconds 3.14 seconds and it actually you know I don't want I want to make the point T does not have to be time and we don't have to be dealing with seconds but I like to think about it that way I like to think about you know maybe this is describing some object in orbit around I don't know something else so now we know the direction as T increased from 0 to PI over 2 to PI we went this way we went counterclockwise so the direction of these parametric equations is in that direction and you might be saying Sal you know why did we just why'd we have to do 3 points we could have just done 2 and made a line if we just had that point in that point you might have immediately said oh we went from there to there but that really wouldn't have been enough because maybe we got from here to there by going the other way around so giving that Third Point lets us know that the direction is definitely counterclockwise and so what happens if we just you know take T from 0 to infinity right what happens if if we bound TT is greater than 0 and less than infinity well we're just going to keep going around this ellipse forever multiple times keep writing over and over to infinite times if we went from if we went from minus infinity to infinity then we would have always been doing it I guess is the way to put it or if we just wanted to trace this out once we could go from T is less than or equal to or T is greater than or equal to zero all the way to T is less than or equal to 2 pi and in this situation t really is the angle that we're tracing out if we were to think of this in polar coordinates this is T at any given time and I just thought I would throw that out there it isn't always but in this case it really is so when you do when you go from 0 to 2 pi radians you've made one circle but this is about parametric equations and not trigonometry so I don't want to focus too much on that but anyway that was neat by we started with this if I just showed you those parametric equations you would have no idea what that looks like but by recognizing the trig identity we were able to simplify it to an ellipse draw the ellipse and then by plotting a couple of points we were able to figure out the direction at which if this was describing a particle in motion the direction in which that particle is actually moving anyway hope you enjoyed that