1. **Problem statement:** Sketch the curve defined by the parametric equations $$x(t) = 2 \cos(t)$$ and $$y(t) = 3 \sin(t)$$ for $$0 \leq t \leq 2\pi$$. Indicate the direction of motion and label at least 3 points with their corresponding $$t$$ values. 2. **Formula and explanation:** The parametric equations describe an ellipse centered at the origin. The general form of an ellipse is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ where $$a$$ and $$b$$ are the semi-major and semi-minor axes. 3. **Convert parametric to Cartesian form:** Using $$x = 2 \cos(t)$$ and $$y = 3 \sin(t)$$, we have $$\frac{x}{2} = \cos(t), \quad \frac{y}{3} = \sin(t)$$ Squaring and adding, $$\left(\frac{x}{2}\right)^2 + \left(\frac{y}{3}\right)^2 = \cos^2(t) + \sin^2(t) = 1$$ This confirms the curve is an ellipse. 4. **Direction of motion:** As $$t$$ increases from 0 to $$2\pi$$, $$x(t)$$ and $$y(t)$$ trace the ellipse counterclockwise because $$\cos(t)$$ starts at 1 and $$\sin(t)$$ starts at 0. 5. **Label points:** - At $$t=0$$: $$x=2\cos(0)=2$$, $$y=3\sin(0)=0$$, point is (2,0). - At $$t=\frac{\pi}{2}$$: $$x=2\cos(\frac{\pi}{2})=0$$, $$y=3\sin(\frac{\pi}{2})=3$$, point is (0,3). - At $$t=\pi$$: $$x=2\cos(\pi)=-2$$, $$y=3\sin(\pi)=0$$, point is (-2,0). 6. **Summary:** The curve is an ellipse centered at the origin with semi-major axis 3 along the y-axis and semi-minor axis 2 along the x-axis, traced counterclockwise from (2,0) at $$t=0$$. **Final answer:** The parametric curve is an ellipse $$\frac{x^2}{4} + \frac{y^2}{9} = 1$$ traced counterclockwise from $$t=0$$ to $$2\pi$$ with labeled points (2,0), (0,3), and (-2,0).