Problem 24 - Calculus Early Transcendentals

Question

\(21-26\) Find a polar equation for the curve represented by the given Cartesian equation. $$x+y=9$$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{9}{\cos \theta + \sin \theta} \).

Step 1: Understand the Polar Coordinate System

In the polar coordinate system, a point is defined by the distance from the origin, called the radius \( r \), and an angle \( \theta \) measured from the positive x-axis. The relationships between Cartesian and polar coordinates are given by \( x = r \cos\theta \) and \( y = r \sin\theta \).

Step 2: Substitute Cartesian Variables with Polar Equivalents

Given the equation \( x+y=9 \), substitute \( x \) and \( y \) with their polar equivalents: \( x = r \cos \theta \) and \( y = r \sin \theta \). This gives us the equation \( r \cos \theta + r \sin \theta = 9 \).

Step 3: Factor Out the Common Factor

Factor out the common factor \( r \) from the left-hand side of the equation: \( r(\cos \theta + \sin \theta) = 9 \).

Step 4: Solve for r

To find \( r \), divide both sides by \( \cos \theta + \sin \theta \): \( r = \frac{9}{\cos \theta + \sin \theta} \). Thus, the polar equation is \( r = \frac{9}{\cos \theta + \sin \theta} \).

Key Concepts

Cartesian to Polar Conversion

Converting between Cartesian and polar coordinates allows us to view mathematical problems from different perspectives. In the Cartesian system, points are described in terms of x and y coordinates. However, polar coordinates express points in terms of radius \( r \) and angle \( \theta \). Transitioning between these systems can make certain types of equations easier to understand and solve.

To convert from Cartesian to polar coordinates, use the relationships:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( r = \sqrt{x^2 + y^2} \)
• \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \)

These transformations are derived from the right triangle relationships and trigonometric identities. By substituting \( x \) and \( y \) in terms of \( r \) and \( \theta \), you can express Cartesian equations in polar terms, as demonstrated in the example of turning the line equation \( x + y = 9 \) into a polar equation.

Polar Coordinate System

The polar coordinate system provides an alternative way to map points in the plane. Unlike Cartesian coordinates that use perpendicular axes, polar coordinates employ a radius and an angle to define locations.

Key components of this system include:

• Radius \( r \): The distance from the point to the origin (the center of the coordinate system).
• Angle \( \theta \): The angle from the positive x-axis to the line connecting the origin with the point, usually measured in radians.

This approach is particularly beneficial for circular and angular motion problems. Instead of manipulating x and y directly, the data is presented in terms of how far and in which direction to move from the origin. This can simplify many mathematical computations, making them more intuitive and easier to visualize in problems involving rotation and circular paths.

Factorization in Algebra

Factorization is a powerful algebraic technique used to simplify equations or expressions by breaking them down into multiples of simpler expressions. In the context of this problem, factorization helps tap into common elements of an equation and simplify the solving process.

When presented with the equation \( r \cos \theta + r \sin \theta = 9 \), recognizing the common factor \( r \) in each term allows us to simplify the equation to \( r(\cos \theta + \sin \theta) = 9 \). By factoring out \( r \), we consolidate the terms and simplify further algebraic manipulation, ultimately solving for \( r \) using division.

Factorization is not only applicable to numerical expressions but also to algebraic terms and functions. It helps reveal insights into the structure and relationships within an equation, often leading to more straightforward solutions.