Problem 60 - A Graphical Approach to Precalculus with Limits

Question

For each equation, find an equivalent equation in rectangular coordinates. Then graph the result. $$r=2 \cos \theta$$

Step-by-Step Solution

Answer

The equivalent equation in rectangular coordinates is \((x-1)^2 + y^2 = 1\).

Step 1: Understand Polar Coordinates

The given equation is in polar coordinates. In polar coordinates, any point is represented with the radius \(r\) and angle \(\theta\). The task is to convert this equation into rectangular coordinates, which means using \(x\) and \(y\) instead of \(r\) and \(\theta\).

Step 2: Use Conversion Formulas

To convert from polar to rectangular coordinates, we use the relationships: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, \(r^2 = x^2 + y^2\) and \(\cos \theta = \frac{x}{r}\). Replace \(r\) and \(\cos \theta\) in the polar equation.

Step 3: Substitute \(r = 2 \cos \theta\)

Start by rewriting the given polar equation: \(r = 2 \cos \theta\). Substituting \(\cos \theta = \frac{x}{r}\), the equation becomes \(r = 2 \cdot \frac{x}{r}\).

Step 4: Rearrange Equation

Multiply both sides by \(r\) to remove the fraction: \(r^2 = 2x\).

Step 5: Substitute \(r^2 = x^2 + y^2\)

Replace \(r^2\) with \(x^2 + y^2\) from the conversion formulas: \(x^2 + y^2 = 2x\).

Step 6: Rearrange to Standard Form

Rearrange the equation to resemble a circle centered at some point. \(x^2 - 2x + y^2 = 0\). Completing the square for the \(x\)-terms gives \((x-1)^2 - 1\), so the equation becomes \((x-1)^2 + y^2 = 1\).

Step 7: Graph the Result

The resulting equation \((x-1)^2 + y^2 = 1\) represents a circle with center at \((1,0)\) and radius 1. Plot this circle on the coordinate plane.

Key Concepts

Polar Coordinates

Polar coordinates represent a point in a plane using two values: radius and angle. The radius \( r \) measures how far the point is from the origin, while the angle \( \theta \) measures the direction from the positive x-axis to the line connecting the point to the origin. This system is particularly useful when dealing with circular and rotational symmetry, as it aligns naturally with the curves and shapes we see in circular motion.
One of the key reasons to use polar coordinates is the simplicity it offers in situations where objects are naturally aligned in circles or arcs:

• Radius is a direct distance, unlike x and y which are components.
• Angles provide a natural way to describe orientation and direction.

By converting polar coordinates into rectangular (Cartesian) coordinates, we can visualize and solve problems using linear approaches, which often aligns better with the tools and more traditional graphing systems found in algebra.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use two perpendicular axes, usually called the x-axis and y-axis, to locate a point. Each point is described by an ordered pair \( (x, y) \):

• \( x \) represents the horizontal distance from the origin.
• \( y \) represents the vertical distance from the origin.

This system is widely used in various fields like mathematics, physics, and engineering due to its ease of use and clarity in plotting points.
In the rectangular coordinate system, transformations are more straightforward, such as translations, reflections, and rotations. This makes them desirable for performing exact calculations and graphing equations, which isn’t as intuitive in polar coordinates for non-circular shapes.

Coordinate Conversion

Coordinate conversion between polar and rectangular systems involves mathematical formulas that relate these two ways of describing a point. These formulas allow mathematicians and engineers to leverage the strengths of both coordinate systems.
For conversion from polar to rectangular coordinates, use:

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

Additionally, for converting the other way around from rectangular to polar:

• \( r = \sqrt{x^2 + y^2} \)
• \( \theta = \arctan\left(\frac{y}{x}\right) \)

These relationships provide the means to translate the elegant curves of polar equations into the linear terms of rectangular coordinates. When dealing with functions or scenarios that mix linear and circular motion, conversion between these models allows for better analysis and solutions.

Equation Graphing

Equation graphing is the process of visually representing the relationships between variables, allowing us to better understand mathematical relationships and their implications. Graphs can display how a change in one quantity affects another or illuminate the nature of mathematical relationships.
For our exercise, converting the equation \( r = 2 \cos \theta \) from polar to rectangular coordinates results in \((x-1)^2 + y^2 = 1\), which is the equation of a circle centered at \((1,0)\) with a radius of 1.
To graph this equation, follow these steps:

• Identify the center of the circle, \((1,0)\).
• Draw a circle with a radius of 1 unit around this center.

Visualization simplifies understanding, allowing us to see symmetries, trends, and insights into how variables are interconnected. Through graphing, theoretical equations turn into approachable visuals, reinforcing learning and discovery.