Consider the following parametric equations. a. Eliminate...

Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in $x$ and $y$.
b. Describe the curve and indicate the positive orientation.
$$x=1-\sin ^{2} s, y=\cos s ; \pi \leq s \leq 2 \pi$$

Key Concepts

Curve Orientation

Curve orientation refers to the direction in which the curve is traversed as the parameter increases. It is important when determining the positive traversal or direction along the curve, which can affect applications like computing integrals along the path. Understanding how the parameter maps to the movement along the curve enables one to identify the starting and ending points, and whether the movement is in a clockwise or counterclockwise direction.

Analyzing the Shape of a Curve

Once the parameter is eliminated, the resulting Cartesian equation provides insight into the geometric nature of the curve. This includes understanding the type of curve (such as a parabola, ellipse, etc.), its orientation, and its symmetry. The process also involves recognizing any restrictions in the range of the variables that might result from the original domain of the parameter.

Trigonometric Identities

Trigonometric identities such as the Pythagorean identity are crucial in eliminating the parameter when the equations involve sine or cosine functions. Recognizing that the square of the sine function can be expressed in terms of the cosine function (or vice versa) helps in finding a relation between x and y. These identities are fundamental tools in simplifying and rewriting trigonometric expressions during the elimination process.

Eliminating the Parameter

This concept involves rewriting the parametric equations in a way that removes the parameter, creating a direct relationship between the variables. It typically requires solving one of the equations for the parameter and substituting into the other equation, or identifying identities or relationships that allow the substitution directly. This process converts the representation into its Cartesian form, making it easier to analyze the curve's properties.

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