(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. $$x=\sin \frac{1}{2} \theta, \quad y=\cos \frac{1}{i} \theta, \quad-\pi \leqslant \theta \leqslant \pi$$

(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.
$$x=\sin \frac{1}{2} \theta, \quad y=\cos \frac{1}{i} \theta, \quad-\pi \leqslant \theta \leqslant \pi$$

Key Concepts

Parametric Equations

Parametric equations represent curves by expressing the coordinates as functions of a separate variable (the parameter). They provide a versatile way to describe curves that might be difficult to represent using a single Cartesian equation.

Elimination of the Parameter

Eliminating the parameter involves rewriting the set of parametric equations to remove the parameter, thereby yielding a Cartesian equation. This step transforms the relationship into one solely between x and y, which can simplify analysis and graphing.

Cartesian Equation

A Cartesian equation relates the x and y coordinates of points on a curve without reference to an external parameter. It provides a direct view of the relationship between the variables and is often used to analyze the geometric properties of the curve.

Curve Sketching

Curve sketching involves drawing the graph of the equation by identifying key characteristics such as intercepts, symmetry, and asymptotes. In the context of parametric equations, it requires evaluating the behavior of the curve over the range of the parameter and plotting key points accordingly.

Direction of Parametric Curves

The direction in which a parametric curve is traced is determined by the increasing values of the parameter. This concept is essential for understanding the movement along the curve, and arrows on a sketch indicate the orientation of traversal, which helps in visualizing dynamic processes or motion along the path.

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