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
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.