Video: Finding the Cartesian Equation of a Curve That Is Defined by Two Parametric Equations

Video Transcript

Find the Cartesian equation of the curve defined by the parametric equations 𝑥 equals two plus cos 𝑡 and 𝑦 equals four cos of two 𝑡.

Remember, a Cartesian equation is one which contains only the variables 𝑥 and 𝑦. So we’re going to need to find a way to eliminate our third variable 𝑡 from our parametric equations. And at first glance, it doesn’t seem to be a nice way to do so. But we can begin by recalling some trigonometric identities. We have cos of two 𝑡 in our second parametric equation. And we know that cos of two 𝑡 is equal to two times cos squared 𝑡 minus one. This means we can rewrite our equation for 𝑦 as four times two cos squared 𝑡 minus one.

Next, we’ll look to rearrange our equation for 𝑥 to make cos of 𝑡 the subject. Once we’ve done that, we’ll be able to find an expression for cos squared 𝑡 in terms of 𝑥. We can subtract two from both sides. And we see that 𝑥 minus two equals cos of 𝑡. Then, by squaring both sides of this equation, we find that cos squared 𝑡 is equal to 𝑥 minus two all squared. And so, we’re now able to replace cos squared 𝑡 with 𝑥 minus two squared. That gives us 𝑦 equals four times two times 𝑥 minus two all squared minus one.

We distribute this first pair of parentheses. And we find that 𝑥 minus two all squared is equal to 𝑥 squared minus four 𝑥 plus four. We distribute again by multiplying each of these terms by two and then simplifying: eight minus one is seven. Well, finally, we distribute one more time by multiplying each term of two 𝑥 squared minus eight 𝑥 plus seven by four. And we find the Cartesian equation of the curve defined by parametric equations 𝑥 equals two plus cos 𝑡 and 𝑦 equals four cos of two 𝑡 is 𝑦 equals eight 𝑥 squared minus 32𝑥 plus 28.