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

Figure 7 $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{16}=1$.

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 $t=2\pi$.

641