(a) Eliminate the parameter to find a Cartesian equation of...

(a) Eliminate the parameter to find a Cartesian equation of the curve.
(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.
$$x=\frac{1}{2} \cos \theta, \quad y=2 \sin \theta, \quad 0 \leqslant \theta \leqslant \pi$$

Key Concepts

Curve Orientation and Tracing

The orientation of a parametric curve refers to the direction in which the curve is traced as the parameter increases. This information is valuable when sketching the curve, as it indicates not just the shape of the curve but also the order in which points are encountered. Arrows are used on the graph to show the direction of traversal corresponding to increasing parameter values.

Graphing Parametric Curves

Graphing parametric curves requires plotting the set of points defined by the parametric equations on the Cartesian plane. This includes understanding the shape of the curve, its domain, range, and any symmetries. The Cartesian equation obtained after eliminating the parameter can help in identifying these properties.

Trigonometric Identities

Trigonometric identities, such as the Pythagorean identity (sin²? + cos²? = 1), are crucial in converting parametric equations that involve trigonometric functions into Cartesian equations. By substituting these identities, one can simplify the relationship between x and y and eliminate the parameter.

Elimination of the Parameter

Elimination of the parameter involves expressing the relationship between x and y directly by removing the intermediate variable. This is often achieved by solving one of the parametric equations for the parameter, or by using identities that link the functions of the parameter together, resulting in a Cartesian equation that solely involves x and y.

Parametric Equations

Parametric equations describe a set of points on a curve as functions of one or more independent parameters. In this context, the x and y coordinates are given as separate functions of a parameter, which allows for a more flexible representation of curves compared to the traditional y = f(x) form.

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