So in recent videos, we learned how to eliminate the parameter from parametric equations. We would solve one of the equations for t, plug it into the other, and end up with a familiar equation of, like, a line or a parabola or something like that. Can we use the same strategy when you're given parametric equations involving trig functions? Well, if I were to try this here, I would get something like the inverse cosine of x. And when I plug it into the y equation, I'll get y = sin inverse cosine of x. It turns out to be a mess. In fact, that actually just won't work. So it turns out with these types of problems, we'll need a different strategy. I'm going to show you how to do that in this video, and the basic idea here is we're going to rewrite both of the equations and use a familiar Pythagorean identity that we've seen before. Let me walk you through how to do this. Let's get started here. Alright? So, again, the basic idea here is that we were solving one equation for t when we were just eliminating the parameter for these types of equations and plugging into the other.
So if I were given something like, for example, x = cost and y = 3sint, how would I go about eliminating the parameter? Well, the idea here is that you're going to solve both of the equations for whatever trig function is inside of them. So for example, here we have cosine, here we have sine, but sometimes you may have sine or tangents and secants and things like that. Alright? So what we're going to do here is we're going to solve both of the equations for whatever trig functions are inside of them. So, for example, this is x = cost. All I have to do is just get cost = x, and I've solved that equation for that whatever trig function. Alright? Same thing for y = 3sint. What I'm going to do is I'm going to try to get sint by itself. All I have to do is just move the 3 over to the other side, but I'm going to flip the equation around like this and say that sint = y3. Okay.
So how does this help us eliminate the parameter? I still have ts inside of both of these equations. Well, now what we're going to do here is we're going to relate our trig functions back to a Pythagorean identity. Remember, these are just the equations that involve the squares of a bunch of trig functions and relating them back to 1. So which identity am I going to use here? Well, in my parametric equation, I have cosine and sine. So it makes sense that I'm going to use the one that involves cosine and sine. So remember, this Pythagorean identity says that sint2+cost2=1. But it actually doesn't have to be an angle. In fact, these variables, the thetas, can actually be anything. It could be x or even t. So what I'm going to do is I'm going to take this first one, and I'm going to rewrite it slightly. So what I'm going to do is I'm going to say that cost2+sint2=1. As long as your calculator is in radians mode, no matter what you plug into these equations, you'll always actually just end up getting 1. So, again, how does this help us eliminate the parameter? Well, I've got an identity here that says the cost2+sint2 = 1. So what I'm going to have to do here is now that I've solved both of the equations for these trig functions I'm just going to have to square both of them. So, in other words, the cosine squared is just going to be squared and the sine squared is just going to be the expression y3 squared. So now, what happens is every time I see cosine squared, I'm just going to replace it with x2 and every time I see the side of t squared, I'm just going to replace it with y32 and then this equals 1 and then I just have a plus sign over here. So now we can see this equation over here, x2+y32=1, is an equation that no longer involves t, and it's also a pretty familiar equation that we've seen before. This actually ends up being the equation of a familiar conic section, which is an ellipse. You have something like x2+y2+someothernumbershere=1. So this actually ends up being how you eliminate the parameter from these types of equations that involve trig functions. You have to solve both of the equations for whatever trig functions are in them, and then you have to relate it back to a familiar Pythagorean identity to get rid of that t variable. That's really all there is to it, folks. So let's go ahead and take a look at some practice problems. Thanks for watching.