Question Video: Sketching a Curve That Is Defined by Two Parametric Equations Mathematics • Higher Education

Video Transcript

Sketch the curve defined by the parametric equations 𝑥 equals two cos 𝑡 minus cos two 𝑡 and 𝑦 equals two sin 𝑡 minus sin two 𝑡, where 𝑡 is greater than or equal to zero and less than or equal to two 𝜋.

Here we’ve been given a pair of parametric equations and asked to sketch the curve over the closed interval for 𝑡 from zero to two 𝜋. We’ll begin by finding some coordinate pairs which satisfy our parametric equations. And since we’re sketching the curve this time rather than just identifying the graph, I’ve chosen values of 𝑡 at subintervals of 𝜋 by four radians. So that’s zero, 𝜋 by four, 𝜋 by two, three 𝜋 by four, all the way through to two 𝜋.

We begin by substituting 𝑡 equals zero into 𝑥. That’s two cos of zero minus cos of two times zero, which is equal to one. We then substitute zero into the equation for 𝑦. That’s two sin of zero minus sin of two times zero, which is zero. By substituting 𝑡 equals 𝜋 by four into 𝑥, we get two cos of 𝜋 by four minus cos of two times 𝜋 by four. And two times 𝜋 by four is 𝜋 by two. This is simply root two.

Repeating this process for 𝑦, we get two sin of 𝜋 by four minus sin of 𝜋 by two, which is minus one plus root two. In fact, it will be easier to plot these if they’re in decimal form. So rounded to three decimal places, we have 1.414 and 0.414. The remaining coordinate pairs that we’re interested in are one, two; negative 1.414, 2.414; negative three, zero; and so on.

Plotting these on a pair of coordinate axes gives us something that looks a little like this. We then join the points as shown and then add arrows to show the direction in which the curve is traced. Remember, we do this by following the values of 𝑡 from smallest to largest. And we sketched the curve defined by the parametric equations 𝑥 equals two cos 𝑡 minus cos two 𝑡 and 𝑦 equals two sin 𝑡 minus sin two 𝑡. This curve has a special name. It’s called a cardioid.