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

Find the Cartesian equation of the curve defined by the parametric equations π‘₯ = 2 + cos 𝑑 and 𝑦 = 4 cos 2𝑑.

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Video Transcript

Find the Cartesian equation of the curve defined by the parametric equations π‘₯ equals two plus cos 𝑑 and 𝑦 equals four cos of two 𝑑.

Remember, a Cartesian equation is one which contains only the variables π‘₯ and 𝑦. So we’re going to need to find a way to eliminate our third variable 𝑑 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 𝑑 in our second parametric equation. And we know that cos of two 𝑑 is equal to two times cos squared 𝑑 minus one. This means we can rewrite our equation for 𝑦 as four times two cos squared 𝑑 minus one.

Next, we’ll look to rearrange our equation for π‘₯ to make cos of 𝑑 the subject. Once we’ve done that, we’ll be able to find an expression for cos squared 𝑑 in terms of π‘₯. We can subtract two from both sides. And we see that π‘₯ minus two equals cos of 𝑑. Then, by squaring both sides of this equation, we find that cos squared 𝑑 is equal to π‘₯ minus two all squared. And so, we’re now able to replace cos squared 𝑑 with π‘₯ minus two squared. That gives us 𝑦 equals four times two times π‘₯ minus two all squared minus one.

We distribute this first pair of parentheses. And we find that π‘₯ minus two all squared is equal to π‘₯ squared minus four π‘₯ 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 π‘₯ squared minus eight π‘₯ plus seven by four. And we find the Cartesian equation of the curve defined by parametric equations π‘₯ equals two plus cos 𝑑 and 𝑦 equals four cos of two 𝑑 is 𝑦 equals eight π‘₯ squared minus 32π‘₯ plus 28.

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