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# Describe the motion of a particle with position $(x, y)$ as $t$ varies in a given interval.$x = 5\sin t$, $\; y = 2\cos t$, $\; -\pi \leqslant t \leqslant 5\pi$

## $x=5 \sin t, y=2 \cos t \Rightarrow \sin t=\frac{x}{5}, \cos t=\frac{y}{2} \cdot \sin ^{2} t+\cos ^{2} t=1 \Rightarrow\left(\frac{x}{5}\right)^{2}+\left(\frac{y}{2}\right)^{2}=1 .$ The motion of theparticle takes place on an ellipse centered at $(0,0) .$ As $t$ goes from $-\pi$ to $5 \pi,$ the particle starts at the point (0,-2) and movesclockwise around the ellipse 3 times.

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Parametric Equations

Polar Coordinates

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