Using Time as the Parameter in Parametric Equations

Describe the motion of an object that moves along a curve so that at time \(t\) it has coordinates \[ \begin{equation*} x( t) =3\cos t\qquad y( t) =4\sin t \qquad 0\leq t\leq 2\pi \end{equation*} \]

Solution We eliminate the parameter \(t\) using the Pythagorean Identity \(\cos ^{2}t+\sin ^{2}t=1\). \[ \begin{equation*} \frac{x^{2}}{9}+\frac{y^{2}}{16}=1\qquad \color{#0066A7}{\cos t= \dfrac{x}{3}, \sin t= \dfrac{y}{4}} \end{equation*} \]

The plane curve is the ellipse shown in Figure 7. When \(t=0\), the object is at the point \((3,0)\). As \(t\) increases, the object moves around the ellipse in a counterclockwise direction, reaching the point \((0,4)\) when \(t=\dfrac{ \pi }{2}\), the point \((-3,0)\) when \(t=\pi\), the point \((0,-4)\) when \(t= \dfrac{3\pi }{2}\), and returning to its starting point \(\left( 3,0\right)\) when \(t=2\pi\).

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