15–30. Working with parametric equations 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 = cos t, y = sin² t; 0 ≤ t ≤ π

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Start with the given parametric equations: \(x = \cos t\) and \(y = \sin^{2} t\), where \(0 \leq t \leq \pi\).

Recall the Pythagorean identity: \(\sin^{2} t + \cos^{2} t = 1\). Use this to express \(\sin^{2} t\) in terms of \(\cos t\).

Since \(y = \sin^{2} t\), rewrite it as \(y = 1 - \cos^{2} t\). Substitute \(x = \cos t\) into this to eliminate the parameter \(t\).

This substitution gives the Cartesian equation relating \(x\) and \(y\): \(y = 1 - x^{2}\).

To describe the curve, recognize that \(y = 1 - x^{2}\) is a downward-opening parabola. The parameter range \(0 \leq t \leq \pi\) corresponds to \(x\) values from \(1\) to \(-1\). The positive orientation follows the direction of increasing \(t\), which moves from \(x=1\) to \(x=-1\) along the curve.

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주요 개념

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Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves and motions.

Eliminating the Parameter

Eliminating the parameter involves manipulating the parametric equations to find a direct relationship between x and y. This often requires using trigonometric identities or algebraic techniques to remove t and express y explicitly or implicitly in terms of x.

Curve Orientation and Description

The orientation of a parametric curve indicates the direction in which the curve is traced as the parameter increases. Describing the curve involves identifying its shape (e.g., circle, ellipse) and specifying the interval of t to understand the portion and direction of the curve.

15–30. Working with parametric equations 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 = cos t, y = sin² t; 0 ≤ t ≤ π