Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

a parametric representation of a curve is given. $$ x=-2 \sin r, y=-3 \cos r ; 0 \leq r \leq 4 \pi $$

Step-by-Step Solution

Answer

The curve is an ellipse with equation \(\frac{x^2}{4} + \frac{y^2}{9} = 1\).

Step 1: Identify Parametric Equations

We are given the parametric equations \(x = -2 \sin r\) and \(y = -3 \cos r\). The parameter \(r\) varies from \(0\) to \(4\pi\).

Step 2: Eliminate the Parameter

To eliminate the parameter \(r\), solve the equations in terms of trigonometric identities. We know that \(\sin^2 r + \cos^2 r = 1\). Express \(\sin r\) and \(\cos r\) from \(x\) and \(y\): \[ \sin r = -\frac{x}{2}, \quad \cos r = -\frac{y}{3}. \]

Step 3: Substitute and Solve for Equation

Substitute \(\sin r\) and \(\cos r\) into the identity \(\sin^2 r + \cos^2 r = 1\): \[ \left(-\frac{x}{2}\right)^2 + \left(-\frac{y}{3}\right)^2 = 1. \] Simplify to get the equation of the curve: \[ \frac{x^2}{4} + \frac{y^2}{9} = 1. \]

Step 4: Recognize the Type of Curve

The equation \(\frac{x^2}{4} + \frac{y^2}{9} = 1\) corresponds to an ellipse centered at the origin. The semi-axis lengths are 2 and 3, making it stretched along the y-axis.

Key Concepts

Trigonometric Identities

Trigonometric identities are powerful tools in mathematics that help in manipulating and simplifying expressions involving trigonometric functions. One fundamental identity is the Pythagorean identity, which states that for any angle \( r \),

• \( \sin^2 r + \cos^2 r = 1 \)

This identity is incredibly useful because it forms the backbone for eliminating parameters in the context of parametric equations.
When given equations in terms of \( \sin r \) and \( \cos r \), we can use this identity to express relationships between \( x \) and \( y \) without the parameter \( r \). In our exercise, identifying \( \sin r = -\frac{x}{2} \) and \( \cos r = -\frac{y}{3} \) allows us to substitute into the identity and eliminate the parameter.
This approach reveals the underlying geometric shape, as in this case, an ellipse.

Ellipse Equation

The ellipse is a type of conic section that can be represented by a standard equation. In the context of our exercise, the ellipse's equation is derived from the parametric forms of \( x \) and \( y \). Once we eliminate the parameter \( r \) using trigonometric identities, we arrive at the standard ellipse equation:

• \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \)

This equation tells us that the ellipse has a horizontal semi-major axis of length 2 and a vertical semi-major axis of length 3.
The equation is derived from scaling the unit circle by different factors along the \( x \) and \( y \) axes. The denominators \( 4 \) and \( 9 \) in the equation reflect these scaling factors, indicating how the circle has been stretched into an ellipse.
Understanding the ellipse equation allows us to visualize the ellipse's orientation and dimensions in a Cartesian plane.

Curve Elimination

In mathematics, curve elimination is the process of removing the parameter from parametric equations to simplify the curve into a more recognizable form. For our exercise, we used curve elimination to transform the parametric equations \( x = -2 \sin r \) and \( y = -3 \cos r \) into a single Cartesian equation.
This technique is particularly useful for analyzing and understanding the shape and properties of the curve. To eliminate the parameter, we express \( \sin r \) and \( \cos r \) in terms of \( x \) and \( y \), then apply the trigonometric identity \( \sin^2 r + \cos^2 r = 1 \).

• Solved for \( \sin r \) and \( \cos r \): \( \sin r = -\frac{x}{2}, \cos r = -\frac{y}{3} \)
• Substitute these into the identity: \( \left(-\frac{x}{2}\right)^2 + \left(-\frac{y}{3}\right)^2 = 1 \)
• Simplify to find the ellipse equation: \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \)

This process reveals that the given parametric equations describe an ellipse, helping us understand the geometric structure it represents. Curve elimination exemplifies how mathematical identities and simplification techniques can transform complex expressions into more digestible forms.