Problem 65 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The equivalent rectangular equation is \(x = 2\) and it represents a vertical line at \(x = 2\).

Step 1: Understand the Polar to Rectangular Conversion

Begin by recalling that in polar coordinates, any point is expressed as \(r, \theta\), where \(r\) is the radius, and \(\theta\) is the angle from the positive x-axis. Rectangular coordinates (Cartesian), use \(x,y\). The formulas for conversion are: \(x = r \cos \theta\) and \(y = r \sin \theta\). Furthermore, \(\sec \theta = \frac{1}{\cos \theta}\).

Step 2: Substitute and Simplify

Given \(r = 2 \sec \theta\), substitute \(\sec \theta = \frac{1}{\cos \theta}\) to rewrite as \(r = \frac{2}{\cos \theta}\). To clear the fraction, multiply both sides by \(\cos \theta\) to obtain \(r \cos \theta = 2\).

Step 3: Convert to Rectangular Coordinates

Using the relationship \(x = r \cos \theta\), replace \(r \cos \theta\) in the equation with \(x\). This substitution gives you \(x = 2\), which is the equation in rectangular coordinates.

Step 4: Graph the Equation

In the rectangular (Cartesian) coordinate system, the equation \(x = 2\) represents a vertical line that crosses the x-axis at x = 2. Plot this vertical line on the graph.

Key Concepts

Rectangular Coordinates

In mathematics, we often use rectangular coordinates, also known as Cartesian coordinates, to describe the position of a point or a set of points in a plane. These coordinates use two values:

• The x-coordinate which describes the horizontal position relative to a vertical axis, known as the y-axis.
• The y-coordinate which describes the vertical position relative to a horizontal axis, known as the x-axis.

Each point is denoted as \((x, y)\) in a two-dimensional plane.
To convert from polar coordinates to rectangular coordinates, you can use the equations:

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

These formulae stem from the trigonometric relationships in a right triangle formed by drawing a perpendicular from the point to the x-axis. By using these conversion formulas, we effectively translate the circular grid used in polar coordinates to a rectilinear grid, facilitating easier graphing and analysis in Cartesian space.

Polar Coordinates

Polar coordinates are another method to describe the location of points on a plane, particularly useful for scenarios involving circular motions or rotations. In this system, we specify a point's location using:

• The radius \(r\), which signifies the distance from the origin to the point.
• The angle \(\theta\), measured in radians or degrees from the positive x-axis.

Polar coordinates are particularly useful in scenarios where systems exhibit radial symmetry or when dealing with periodic phenomena.
Conversion to rectangular coordinates is often necessary to utilize the simplicity of Cartesian grids in calculations. For instance, using the

• \(\sec \theta = \frac{1}{\cos \theta}\), in polar coordinates transforms the radial distance to align with the x-axis movement.
• From the given equation \(r = 2 \sec \theta\), we can manipulate it to form \(r \cos \theta = 2\), paving the way for conversion to rectangular coordinates.

Being adept in switching between polar and rectangular coordinates empowers us to address a wider range of mathematical and practical problems effectively.

Graphing Equations

Graphing equations is a crucial part of understanding functions visually and their relationships in a mathematics class. When we graph them using rectangular coordinates, it becomes easier to interpret slopes, intercepts, and other features visually.
To graph an equation, like \(x = 2\), in rectangular coordinates, means to plot all the points that satisfy the condition of the equation. For our solution:

• \x = 2 represents a vertical line on a coordinate plane that intersects the x-axis at x = 2.
• Every point along this line has an x-coordinate of 2, regardless of its y-coordinate.

The graph visually represents solutions of the equation, providing an intuitive understanding of concepts and relationships.
This graphical representation allows one to see how equations behave, which is essential not just in solving equations but also in various fields such as physics, engineering, and economics.