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

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

Key Concepts

Direction of Traversal in Parametric Curves

In a parametric context, the parameter not only traces out the points of the curve but also determines the direction in which the curve is traversed. By analyzing how x and y change as the parameter increases, one can indicate the movement along the curve, which is critical for understanding the orientation and behavior of paths in applications such as physics and computer graphics.

Parametric Representation of Curves

Parametric equations express the coordinates of the points on a curve as functions of a parameter, offering a flexible way to represent curves that might be difficult to describe with a single explicit Cartesian equation. This approach is particularly useful in capturing the dynamics and structure of curves as the parameter varies.

Parameter Elimination

This concept involves removing the parameter from a set of parametric equations to derive a direct relationship between the Cartesian variables. It often relies on algebraic manipulations or identities, allowing one to obtain an equation in x and y that represents the same curve as the original parametrization.

Trigonometric Identities

Trigonometric identities, such as sin²(?) + cos²(?) = 1, are essential tools for eliminating parameters in curves defined by sine and cosine functions. These identities help simplify the expressions and reveal the inherent geometric structure of the curve, such as its being part of a circle or other conic section.

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