Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding Parametric Equations

Parametric equations describe a set of related quantities as functions of an independent parameter. Here, the parameter is \( r \), and for each value of \( r \), we get a point \((x, y)\) on the curve where \(x = 3\sin r\) and \(y = -2\cos r\).

Step 2: Identifying the Shape

The given parametric equations \(x = 3\sin r\) and \(y = -2\cos r\) resemble the structure of parametric forms of ellipses. Recall that the general form is \(x = a\cos t\) and \(y = b\sin t\) for an ellipse centered at the origin. However, here we have a variation based on sine and cosine.

Step 3: Expressing the Parametric Equations in Cartesian Form

Convert the parametric equation to eliminate the parameter \(r\). Use the fact that \(\sin^2 r + \cos^2 r = 1\). Hence, \(\frac{x}{3} = \sin r\) and \(\frac{y}{-2} = \cos r\). So, \(\left(\frac{x}{3}\right)^2 + \left(\frac{y}{-2}\right)^2 = \sin^2 r + \cos^2 r = 1\), resulting in the equation \(\frac{x^2}{9} + \frac{y^2}{4} = 1\).

Step 4: Analyzing the Cartesian Equation

The equation \(\frac{x^2}{9} + \frac{y^2}{4} = 1\) confirms that the curve is an ellipse. It has its major axis along the y-axis with a length of 4 (2 times the distance from the origin to the "vertex" \(x = 0, y = -2\) or \(x = 0, y = 2\)) and its minor axis along the x-axis with a length of 6 (2 times the distance from the origin to the "vertex" \(x = -3, y = 0\) or \(x = 3, y = 0\)).

Step 5: Verifying the Range

The parameter \(r\) ranges from 0 to \(2\pi\). This ensures the tracing of the full ellipse once in the counterclockwise direction.

Key Concepts

The Geometry of an Ellipse

An ellipse is a fascinating geometric shape defined by specific mathematical properties. Imagine stretching a circle along one axis, which creates an ellipse. This shape has two axes: the major axis and the minor axis.

A standard form of an ellipse's equation is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes, respectively. The ellipse is centered at the origin (0,0).

The major axis is the longest diameter, while the minor axis is the shortest. In our parametric equation for the ellipse, \(x = 3\sin r\) and \(y = -2\cos r\), we detect the semi-major axis length as 3, due to the maximum value of the sine function, and the semi-minor axis length as 2, from the cosine function. This gives us an ellipse rotated in its orientation compared to its standard position.

To sum it up:

• Major Axis: The longest diameter (along the y-axis in our solution)
• Minor Axis: The shortest diameter (along the x-axis in our solution)
• Center: At the origin

Understanding this helps one visualize how an ellipse looks and its orientation based on the mathematical expressions.

Understanding Cartesian Coordinates

Cartesian coordinates provide a way to specify the location of a point on a plane using two numbers: \((x, y)\). These numbers correspond to the point's horizontal and vertical position, respectively.

A Cartesian coordinate system is composed of two perpendicular axes, typically labeled as the x-axis (horizontal) and y-axis (vertical). The point where these axes intersect is the origin, denoted as \((0, 0)\). Each point on the plane has a unique pair of coordinates.

When converting a parametric equation to a Cartesian equation, we eliminate the parameter, in our case \( r \). By using trigonometric identities, we can express the entire graph of the curve in terms of x and y. For example, given the parametric equations \( x = 3 \sin r \) and \( y = -2 \cos r \), we have derived the Cartesian form as \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \).

• Coordinates: Expressed as \((x, y)\)
• Axes: x-axis (horizontal) & y-axis (vertical)
• Origin: Point (0, 0)

This system allows us to represent curves like the ellipse effectively as mathematical equations.

Exploring Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any values of the involved variables. They are fundamental tools in mathematics, reducing complex expressions and relating different trigonometric functions.

A primary identity used often is \( \sin^2 r + \cos^2 r = 1 \). This identity connects the sine and cosine of the same angle and is crucial in transforming parametric equations into Cartesian coordinates. In the current example, given \( x = 3 \sin r \) and \( y = -2 \cos r \), we use this identity to deduce that \( \left(\frac{x}{3}\right)^2 + \left(\frac{y}{-2}\right)^2 = 1 \).

• Basic Identity: \( \sin^2 r + \cos^2 r = 1 \)
• Purpose: Relates sine and cosine functions
• Application: Used to derive the Cartesian form from parametric equations

By mastering these identities, one can manipulate and transform trigonometric equations with confidence and insight.