91Ó°ÊÓ

A Cartesian equation represents a single equation linking the x and y variables without any parameters. This form is particularly useful for plotting points on a plane because it directly shows the relationship between x and y. In contrast, parametric equations involve a third variable, often a parameter like \( \phi \). To find the Cartesian equation from parametric ones, the parameter must be eliminated. This is done by deriving expressions for x and y separately in terms of the parameter and then using algebraic identities or transformations that link these expressions together. In our exercise, transforming \( x = 4 \cos \phi \) and \( y = 1 - \sin \phi \) into a Cartesian equation involved substituting into the Pythagorean identity.

An ellipse is a geometric shape that appears as an elongated circle. It is defined mathematically by an equation that is a generalization of the circle equation. The canonical form of an ellipse equation is \( \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \), where \((h, k)\) is the center of the ellipse, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis. In our specific problem, we derived the equation \( \frac{x^2}{16} + (y - 1)^2 = 1 \), which indicates an ellipse centered at \((0, 1)\). Here, the semi-major axis length is 4 (horizontal) and the semi-minor axis length is 1 (vertical). This transformation from the parametric form reveals the symmetric properties of the shape, where the lengths of the axes determine how stretched it appears.

The Pythagorean identity is a fundamental trigonometric relation stating that \( \cos^2 \phi + \sin^2 \phi = 1 \). This identity stems from the Pythagorean theorem and applies to any angle \( \phi \). It is especially powerful when converting between parametric and Cartesian equations because it offers a direct way to eliminate the parameter. In this exercise, we used the identity to transform the parametric equations \( x = 4 \cos \phi \) and \( y = 1 - \sin \phi \) into the Cartesian form. After expressing \( \cos \phi \) as \( \frac{x}{4} \) and \( \sin \phi \) as \( 1 - y \), substituting them into the Pythagorean identity allowed us to connect x and y in a single equation without \( \phi \). This step is crucial in linking trigonometric relationships to algebraic expressions.

Completing the square is a useful algebraic technique to simplify quadratic equations and express them in a form that makes the properties of the equation more obvious. For a quadratic expression in y, such as \( y^2 - 2y \), completing the square involves rewriting it as \((y - 1)^2 - 1\). This transformation is crucial for our problem as it helped adjust the ellipse equation to its standard form. By completing the square, we moved from \( y^2 - 2y \) to \( (y - 1)^2 \), simplifying the equation to \( \frac{x^2}{16} + (y - 1)^2 = 1 \). Such a transformation clearly highlights the shifted center of the ellipse and makes further calculations or graphical interpretations straightforward. Understanding this technique can be pivotal for solving a wide array of problems, particularly in geometry and calculus.