Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 17-22, find the Cartesian equations of the graphs of the given polar equations. \(r=3\)

Step-by-Step Solution

Answer

The Cartesian equation is \( x^2 + y^2 = 9 \).

Step 1: Understanding the Problem

We need to convert the given polar equation into a Cartesian equation. The given polar equation is \( r = 3 \). Our goal is to find an equivalent equation using Cartesian coordinates \( x \) and \( y \).

Step 2: Identifying the Polar to Cartesian Transformation

To convert from polar to Cartesian coordinates, we use the following relationships: \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). Additionally, the equation \( r^2 = x^2 + y^2 \) links the radius in polar coordinates to Cartesian coordinates.

Step 3: Substituting the Given Polar Equation into Cartesian Variables

Since the polar equation is \( r = 3 \), we substitute \( r \) in the Cartesian transformation formula: \( r^2 = x^2 + y^2 \). Now inserting \( r = 3 \) will give us the equation \( (3)^2 = x^2 + y^2 \).

Step 4: Simplifying the Cartesian Equation

Simplifying \( 3^2 \) gives us 9, leading to the equation \( x^2 + y^2 = 9 \). This represents a circle centered at the origin (0,0) with a radius of 3 on the Cartesian plane.

Key Concepts

Polar to Cartesian Conversion

Polar to Cartesian conversion is a fundamental concept in mathematics that allows us to translate equations from polar coordinates to Cartesian coordinates. The polar coordinate system is defined by a radial distance \( r \) and an angle \( \theta \), while the Cartesian coordinate system uses \( x \) and \( y \) to locate points on the plane.

To perform this transformation, we can use the relationships:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)
• \( r^2 = x^2 + y^2 \)

The first two formulas help translate radial and angular descriptions into the familiar \( x \)- and \( y \)-coordinate form, and the third formula connects both coordinate systems through the radius squared.

Consider the polar equation \( r = 3 \). By applying the formula \( r^2 = x^2 + y^2 \), we derive \( x^2 + y^2 = 9 \). This transformed equation reveals the corresponding shape on the Cartesian plane, helping us visualize and work with concepts intuitively.

Coordinate Transformation

Coordinate transformation is a method used to alter the way we represent points and shapes. The goal is to make analysis or visualization easier by converting the equation from one coordinate system to another. It involves mathematical formulas that bridge the gap between different systems, such as polar to Cartesian conversions.

In a coordinate transformation:

• Identify the original coordinate system and its defining equations or relations.
• Apply the necessary conversion formulas to modify these equations to fit the target coordinate system.
• Analyze the new equation to interpret geometrical features more easily.

For instance, changing from polar to Cartesian coordinates using the equation \( r = 3 \) gives us the Cartesian form \( x^2 + y^2 = 9 \). This process helps simplify complex problems, allowing for easier graphing and application of calculus concepts in a more familiar framework.

Equations of Circles

In Cartesian coordinates, the equation of a circle with a center at the origin is given by \( x^2 + y^2 = r^2 \), where \( r \) is the radius. Circles represent one of the simplest closed shapes, and their equations illustrate a perfect symmetry around their center.

The equation \( x^2 + y^2 = 9 \) describes a circle centered at the origin with a radius of 3. This is derived from the polar equation \( r = 3 \).

Why is understanding the equation of a circle important?

• It helps in visualizing geometric transformations and understanding spatial relationships.
• Circles are fundamental in trigonometry and calculus, often forming the basis for more complex analyses.
• Understanding circle equations aids in solving intersection problems and evaluating motion in physics and engineering.

By mastering these basic circle equations, students can extend their knowledge to encompass ellipses, parabolas, and other curved shapes with similar analytical methods.