Problem 165 - Calculus Volume 3

Question

Convert the polar equation to rectangular form and sketch its graph. $$ r=\theta $$

Step-by-Step Solution

Answer

Rectangular form: \( \sqrt{x^2 + y^2} = \arctan\left(\frac{y}{x}\right) \). It represents a spiral starting from the origin.

Step 1: Introduction to the Problem

We need to convert the polar equation \( r = \theta \) into rectangular form and sketch its graph. In polar coordinates, \( r \) is the distance from the origin, and \( \theta \) is the angle from the positive x-axis.

Step 2: Understanding Polar Coordinates

In polar coordinates, we have \( r \) and \( \theta \) which describe a point in the plane. The relationship with rectangular coordinates (\( x, y \)) is \( x = r \cos \theta \) and \( y = r \sin \theta \). Furthermore, we have the auxiliary equations: \( r^2 = x^2 + y^2 \) and \( \tan \theta = \frac{y}{x} \).

Step 3: Express \( \theta \) in Terms of Rectangular Coordinates

Given the polar equation \( r = \theta \), we first need \( \theta \) in terms of \( x \) and \( y \). From \( \tan \theta = \frac{y}{x} \), we derive \( \theta = \arctan\left(\frac{y}{x}\right) \).

Step 4: Substitute \( \theta \) into the Polar Equation

Replace \( \theta \) in the polar equation \( r = \theta \) with the expression we found: \[ r = \arctan\left(\frac{y}{x}\right) \]

Step 5: Substitute for \( r \) in Rectangular Form

Also remember \( r = \sqrt{x^2 + y^2} \). Substitute this into the equation, giving: \[ \sqrt{x^2 + y^2} = \arctan\left(\frac{y}{x}\right) \] Thus, the rectangular form of the given polar equation is \( \sqrt{x^2 + y^2} = \arctan\left(\frac{y}{x}\right) \).

Step 6: Sketch the Graph

To sketch the graph, note that it represents a spiral. As \( \theta \) (angle) increases, \( r \) (radius) increases, tracing out a spiral pattern outward from the origin. This is because as the angle increases beyond its initial range of \([0, 2\pi] \), \( r \) increases linearly.

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using two values: the radial distance from a fixed point, known as the pole (typically the origin of a coordinate system), and the angle from a fixed direction, usually the positive x-axis. In the polar coordinate system, a point is expressed as \( (r, \theta) \). Here, \( r \) is the distance from the origin, and \( \theta \) is the angular measurement in radians or degrees.

• Distance \( r \): Indicates how far the point is from the origin. It can be positive or zero.
• Angle \( \theta \): Determines the direction of the point from the positive x-axis and this can be any real number.

Polar coordinates are especially useful in scenarios where the relationship between point and problem is more naturally circular or radial. This system simplifies equations and computations involving rotation and spirals.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, express points in terms of two perpendicular axes: x and y. A point in this system is written as \( (x, y) \), where:

• \( x \)-coordinate: Represents the horizontal position along the x-axis.
• \( y \)-coordinate: Indicates the vertical position along the y-axis.

This system is well-suited for problems involving straight lines and is commonly used in algebra and geometry.
To convert from polar to rectangular, use the formulas:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

These transformations enable us to move easily between circular and linear representations of data.

Graph Sketching

Graph sketching involves drawing a visual representation of an equation or function on a coordinate plane. In this case, our goal is to sketch the graph of a polar equation after transforming it into rectangular coordinates.
This process helps us understand the nature and behavior of the function.

• For a spiral equation such as \( r = \theta \), the graph will show an outward spiral, reflecting the increase in \( r \) as \( \theta \) rises.
• Turning this polar equation into a rectangular form reveals a relation involved between \( x \) and \( y \), even if it's more complex.

A good sketch considers the symmetry, curvature, and key points on the graph, providing insights into the overall characteristics of the figure.

Coordinate Transformation

Coordinate transformation is the process of converting one set of coordinates into another. In mathematics, particularly in geometry and related fields, it's crucial for making sense of data in different contexts. Here, we focus on converting polar coordinates to rectangular coordinates, making it easier to handle equations algebraically.
Steps for transforming polar to rectangular include:

• Finding \( \theta \) in terms of \( x \) and \( y \) using the arctan function: \( \theta = \arctan\left(\frac{y}{x}\right) \).
• Substituting \( r=\theta \) and using \( r=\sqrt{x^2+y^2} \) to form the equation \( \sqrt{x^2+y^2} = \arctan\left(\frac{y}{x}\right) \).

This transformed equation provides a path to sketch graphs and solve real-world problems using linear systems.