Problem 154 - Calculus Volume 3

Question

Describe the graph of each polar equation. Confirm each description by converting into a rectangular equation. $$ r=3 $$

Step-by-Step Solution

Answer

The graph is a circle centered at the origin with radius 3.

Step 1: Understand the Polar Equation

The given equation is in polar form: \( r = 3 \). This means that no matter the angle \( \theta \), the radial distance from the origin is always 3. Thus, in polar coordinates, this describes a circle centered at the pole (origin) with a radius of 3.

Step 2: Convert to Rectangular Coordinates

To convert the polar equation \( r = 3 \) into a rectangular equation, use the relationships between polar and rectangular coordinates: \( x = r \cos \theta \) and \( y = r \sin \theta \). Also, \( r = \sqrt{x^2 + y^2} \). So, substitute \( r = 3 \) into this equation: \[ x^2 + y^2 = r^2 = 9 \] Thus, the rectangular form is the equation of a circle centered at the origin with radius 3.

Step 3: Analyze the Rectangular Equation

The rectangular equation \( x^2 + y^2 = 9 \) confirms the description obtained from the polar equation. It represents a circle with a radius of 3 centered at the origin in the Cartesian coordinate system.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a system for representing points on a plane. This system uses pairs of numbers, typically labeled as \( (x, y) \), to describe a point's position based on its horizontal and vertical distances from a reference point, known as the origin. The origin is usually denoted by the coordinates \( (0, 0) \).Rectangular coordinates map the two-dimensional space using grid lines, where:

• \( x \): The horizontal axis, also called the abscissa, provides the left-right position.
• \( y \): The vertical axis, also referred to as the ordinate, gives the up-down position.

Understanding rectangular coordinates is essential for converting equations from or to this format. For example, a circle centered at the origin with radius 3 in rectangular form is described by the equation \( x^2 + y^2 = 9 \). This equation shows that any point fitting this equation lies exactly 3 units away from the origin, forming a perfect circle on the Cartesian plane.

Equation Conversion

Equation conversion involves translating an equation from one coordinate system to another, often to gain insights or simplify analysis. Converting between polar and rectangular coordinates is a common task in mathematics, helping to visualize and analyze geometric shapes more effectively.To convert from polar coordinates \( (r, \theta) \) to rectangular coordinates \( (x, y) \):

• Use the formulas \( x = r \cos \theta \) and \( y = r \sin \theta \). These formulas stem from the trigonometric relationships and define how the angle and radius are related to the horizontal and vertical components.
• Conversely, to convert from rectangular to polar, use \( r = \sqrt{x^2 + y^2} \) and \( \theta = \tan^{-1} \left(\frac{y}{x}\right) \).

In the exercise at hand, converting the polar equation \( r = 3 \) to rectangular coordinates involves recognizing that \( r \) represents the radial distance thus substituting into \( r^2 = x^2 + y^2 \), resulting in the more familiar form of a circle, \( x^2 + y^2 = 9 \). This showcases the power of conversion for visualizing and confirming geometric representations.

Graphs of Polar Equations

Graphs of polar equations provide a distinctive way of visualizing mathematical relationships, especially useful for depicting curves and shapes centered around a point, often the origin. Polar coordinates are represented as \( (r, \theta) \), where \( r \) denotes the distance from the origin, and \( \theta \) represents the angle from the positive x-axis.In polar graphs:

• Consistent distances \( r \) create recognizable shapes like circles. For instance, \( r = 3 \) describes a circle with radius 3 centered at the origin. This is because the radial distance remains constant regardless of the angle.
• More complex equations involving \( \theta \) can produce elegant curves such as spirals or roses, each characterized by cyclical variation in \( r \).

Using polar graphs can simplify understanding and construction of certain shapes which can be awkward in rectangular coordinates, making them invaluable for topics ranging from navigation to advanced geometry. The conversion of polar equations to rectangular forms assists in validating their geometric implications in a different coordinate system.