Given are parametric equations and a parameter interval for the motion of a particle in the​ xy-plane. Identify the​ particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion.
xequals=2 plus sine t2+sint​,    yequals=cosine t minus 3cost−3​,    0less than or equals≤tless than or equals≤piπ

The graph shows the path of the particle as the upper half of a circle with a center at \( (2, -3) \) and a radius of 1. The particle starts at \( (2, -2) \) (marked in green) and moves counterclockwise to \( (2, -4) \) (marked in red).

### Summary:
- **Path:** The upper half of the circle \((x - 2)^2 + (y + 3)^2 = 1\).
- **Portion Traced:** From \( (2, -2) \) to \( (2, -4) \).
- **Direction of Motion:** Counterclockwise.

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### Related Questions:
1. How do we find parametric equations from a given Cartesian equation?
2. What are the parametric representations for standard geometric shapes like circles and ellipses?
3. How does changing the parameter interval affect the path traced by a particle?
4. What happens to the graph if the parameter interval is extended beyond \( [0, \pi] \)?
5. Can we express the path in polar coordinates instead of Cartesian?

**Tip:** When converting parametric equations to Cartesian form, always look for trigonometric identities or algebraic relationships to eliminate the parameter.

Learn how to identify the path of a particle in the xy-plane using parametric equations. We convert parametric equations to Cartesian form, graph the resulting equation, and analyze the particle's motion from start to end. Includes a detailed example with graphical representation.

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