Problem 55 - Thomas Calculus

Question

Replace the Cartesian equations in Exercises \(49-62\) by equivalent polar equations. $$ \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 $$

Step-by-Step Solution

Answer

The polar form is \( r^2 = \frac{1}{\frac{\cos^2(\theta)}{9} + \frac{\sin^2(\theta)}{4}} \).

Step 1: Understanding the Cartesian Equation

The given Cartesian equation is \( \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 \). This represents an ellipse where \(x^2\) is divided by 9 and \(y^2\) is divided by 4.

Step 2: Converting to Polar Coordinates

In polar coordinates, \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \). Our goal is to express the Cartesian equation in terms of \( r \) and \( \theta \).

Step 3: Substituting Polar Expressions

Substitute \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \) into the equation: \[ \frac{(r\cos(\theta))^{2}}{9} + \frac{(r\sin(\theta))^{2}}{4} = 1 \].

Step 4: Simplifying the Equation

Expand the equation: \[ \frac{r^2\cos^2(\theta)}{9} + \frac{r^2\sin^2(\theta)}{4} = 1 \]. Combine terms: \[ r^2 \left( \frac{\cos^2(\theta)}{9} + \frac{\sin^2(\theta)}{4} \right) = 1 \].

Step 5: Factor Out \( r^2 \)

Solve for \( r^2 \): \[ r^2 = \frac{1}{\frac{\cos^2(\theta)}{9} + \frac{\sin^2(\theta)}{4}} \].

Step 6: Final Polar Equation

This equation \( r^2 = \frac{1}{\frac{\cos^2(\theta)}{9} + \frac{\sin^2(\theta)}{4}} \) represents the ellipse in polar form.

Key Concepts

Cartesian Equations

A Cartesian equation is a mathematical expression that uses the Cartesian coordinate system, represented by the axes labeled \( x \) and \( y \). It's a way to describe geometrical shapes and figures on a plane. In this coordinate system, each point is defined by an ordered pair \( (x, y) \).
For example, in the equation \( \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 \), the relationship between \( x \) and \( y \) defines an ellipse. The numbers 9 and 4 are denominators of the squared terms, indicating the spread or radius-like attributes along the \( x \) and \( y \) axes, respectively.

• 9 under \( x^2 \) suggests the ellipse stretches more along the x-axis.


• 4 under \( y^2 \) hints the ellipse stretches less along the y-axis.

These kinds of equations are essential in various fields like physics, engineering, and computer graphics, as they help model real-world behaviors and systems.

Converting Coordinates

Converting between coordinate systems like Cartesian and polar is beneficial to simplify problems and calculations in geometry and physics. In polar coordinates, every point is described using a radius \( r \) and an angle \( \theta \), which can be advantageous when dealing with circles or rotational patterns.
To convert Cartesian coordinates \( (x, y) \) into polar coordinates \( (r, \theta) \):

• Use the relation \( x = r\cos(\theta) \).


• Use the relation \( y = r\sin(\theta) \).

The advantage lies in transforming complex algebraic expressions into simpler trigonometric forms, enabling easier integration or differentiation. In our exercise, substituting the polar expressions into the Cartesian equation helps us represent the ellipse using a different perspective that can be more useful in certain applications.

Ellipse

An ellipse is a stretched circle with two axes – a major axis and a minor axis. In the Cartesian equation \( \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 \), the ellipse's shape is determined by the values under \( x^2 \) and \( y^2 \). Here, 9 and 4 represent the semi-major and semi-minor axes squared, respectively.

• The longer axis (the major axis) determines the furthest distance across the ellipse.


• The shorter axis (the minor axis) is the shortest distance across.

Ellipses are found in many areas of science, such as astronomy where planetary orbits are often elliptical. Understanding how to write the equation of an ellipse in different coordinate systems allows us to analyze these shapes more effectively in various scenarios.

Trigonometric Substitution

Trigonometric substitution is a technique often employed to simplify the integration of functions. It involves substituting trigonometric identities into algebraic expressions, particularly those involving quadratic terms.
In our problem, we use trigonometric identities to rewrite Cartesian coordinates in polar form: substituting \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \) into the original equation. This step makes use of the fundamental trigonometric identities, such as:

• \( \cos^2(\theta) + \sin^2(\theta) = 1 \)

With this method, we simplify the ellipse equation and solve it in the polar coordinate system. Trigonometric substitution is a powerful mathematical tool that provides greater flexibility in solving complex algebraic equations.