Video: Graphing Pairs of Parametric Equations

Video Transcript

Consider the parametric equations 𝑥 of 𝑡 equals two sin 𝑡 and 𝑦 of 𝑡 equals three cos 𝑡, where 𝑡 is greater than zero and less than three 𝜋. Which of the following is the sketch of the given equations?

Here we’ve been given a pair of parametric equations and asked to find the sketch of the curve over the open interval for 𝑡 from zero to three 𝜋. We’re going to need to find some coordinate pairs that satisfy our parametric equations. Now whilst we don’t technically want to include 𝑡 equals zero and 𝑡 equals three 𝜋, we know that 𝑡 approaches both of these values. So we’ll use direct substitution to find the coordinate pair that our curve approaches at the end points on our open interval.

Let’s choose intervals of 𝜋 by two radians. Now if we were actually looking to sketch the curve, we may wish to choose, say, 𝜋 by four as our subinterval. But we’re just simply looking to compare our coordinates to the given graphs. To find the first ordered pair, we substitute 𝑡 equals zero into each of our parametric equations. That gives us 𝑥 equals two sin zero and 𝑦 equals three cos zero. Our first ordered pair is zero, three.

We then substitute 𝑡 equals 𝜋 by two. And we get 𝑥 equals two sin of 𝜋 by two and 𝑦 equals three cos of 𝜋 by two. That gives us 𝑥 equals two and 𝑦 equals zero. We continue the pattern by substituting 𝑡 equals 𝜋 to get two sin 𝜋 for 𝑥 and three cos 𝜋 for 𝑦, giving us an ordered pair of zero, negative three. Substituting 𝑡 equals three 𝜋 by two into each equation, and we get 𝑥 equals negative two and 𝑦 equals zero. For 𝑡 equals two 𝜋, we get the ordered pair zero, three. And our last two ordered pairs occur when 𝑡 equals five 𝜋 by two and three 𝜋. And they are two, zero and zero, negative three, respectively.

Let’s compare these to each of our sketches. And we need to ensure that we move in increasing values of 𝑡. That is, we begin at 𝑡 equals zero and move all the way through to 𝑡 equals three 𝜋. That leaves us with either B or C. And in fact, we move in a clockwise direction. So actually, we’re interested in C. And you might have noticed that the values themselves repeat. It should be quite clear now that, due to the nature of the shape of our curve, this pattern will continue forever. The sketch of the equations 𝑥 of 𝑡 equals two sin 𝑡 and 𝑦 of 𝑡 equals three cos 𝑡 is C.