1. **State the problem:** We are given parametric equations $x = 5 \cos t$ and $y = 2 \sin t$ with parameter $t$ ranging from $0$ to $2\pi$. We want to understand the relationship between $x$ and $y$ and analyze the curve. 2. **Formula and important rules:** The parametric equations describe a curve in the plane. To find the Cartesian equation, we use the Pythagorean identity $\sin^2 t + \cos^2 t = 1$. 3. **Express $\cos t$ and $\sin t$ in terms of $x$ and $y$:** $$\cos t = \frac{x}{5}, \quad \sin t = \frac{y}{2}$$ 4. **Use the Pythagorean identity:** $$\left(\frac{x}{5}\right)^2 + \left(\frac{y}{2}\right)^2 = 1$$ 5. **Simplify the equation:** $$\frac{x^2}{25} + \frac{y^2}{4} = 1$$ 6. **Interpretation:** This is the equation of an ellipse centered at the origin with semi-major axis 5 along the $x$-axis and semi-minor axis 2 along the $y$-axis. 7. **Evaluate points from the table:** For each $t$ value, compute $x$ and $y$: - At $t=0$: $x=5\cos 0=5$, $y=2\sin 0=0$ - At $t=\frac{\pi}{4}$: $x=5\cos \frac{\pi}{4}=\frac{5\sqrt{2}}{2}$, $y=2\sin \frac{\pi}{4}=\sqrt{2}$ - At $t=\frac{\pi}{2}$: $x=5\cos \frac{\pi}{2}=0$, $y=2\sin \frac{\pi}{2}=2$ - And so on for other $t$ values. This matches the points plotted on the graph. **Final answer:** The parametric equations describe the ellipse $$\frac{x^2}{25} + \frac{y^2}{4} = 1$$