Problem 40 - Precalculus with Calculus Previews

Question

Find a polar equation that has the same graph as the given rectangular equation. $$ x+1=0 $$

Step-by-Step Solution

Answer

The polar equation is \( r \cos \theta = -1 \).

Step 1: Understand the Rectangular Equation

The given rectangular equation is \( x + 1 = 0 \). This equation represents a vertical line in the Cartesian coordinate system, specifically the line \( x = -1 \).

Step 2: Identify Polar Coordinate System Relations

Recall that in polar coordinates, the relationship between the Cartesian coordinates \( (x, y) \) and polar coordinates \( (r, \theta) \) is given by \( x = r \cos \theta \) and \( y = r \sin \theta \).

Step 3: Substitute into the Rectangular Equation

Substitute \( x = r \cos \theta \) into the rectangular equation \( x = -1 \). This gives us the equation:\[ r \cos \theta = -1 \]

Step 4: Formulate the Polar Equation

The polar equation that represents the same graph as the rectangular equation \( x + 1 = 0 \) is thus:\[ r \cos \theta = -1 \]

Key Concepts

Cartesian Coordinate System

When working with equations in mathematics, it often involves the Cartesian coordinate system. This is a familiar way to display points in a two-dimensional space, using a pair of numerical values: \( x \) and \( y \). These values represent the horizontal and vertical distances from the origin, respectively. The entire system can be visualized as a grid with two perpendicular axes, the x-axis (horizontal) and the y-axis (vertical).

• The Cartesian plane is named after the mathematician René Descartes, who was pivotal in its development.
• Each point on the plane corresponds to a unique pair of coordinates \((x, y)\).
• Equations formed in this system typically express relationships between \( x \) and \( y \).

For example, the equation \( x + 1 = 0 \) describes a specific line: it indicates all points on the line where \( x = -1 \), regardless of the value of \( y \). This is known as a vertical line on the Cartesian plane. It's one of the simplest forms of a linear equation, represented by a straight line.

Rectangular Equation

A rectangular equation expresses a relationship between \( x \) and \( y \) within the Cartesian coordinate system. These are equations that involve only the standard coordinate axes. They can describe a wide variety of geometrical entities such as lines, circles, ellipses, and more complex shapes depending on their form.A simple example would be the linear equation \( x + 1 = 0 \). Let's break it down:- **Understanding the format:** Standard form for a line is \( ax + by = c \). Here, \( a = 1 \), \( b = 0 \), and \( c = -1 \). - **Horizontal vs. Vertical Lines:** If an equation involves only \( x \) with no \( y \), it's a vertical line, and similarly, if only \( y \) is involved without \( x \), it's a horizontal line.Though simple, rectangular equations are crucial in many fields of science and engineering. They facilitate the translation of abstract mathematical ideas into visually discernible forms on a graph.

Polar Coordinates

Polar coordinates offer an alternative to the Cartesian coordinate system and are particularly useful in certain situations. Instead of using \( x \) and \( y \) to mark a point, polar coordinates use \( r \) and \( \theta \).

• \( r \) represents the radial distance from the origin (the center point), essentially how far away a point is in the direction of \( \theta \).
• \( \theta \) stands for the angle from the positive x-axis, usually measured in radians.

The transformation between rectangular and polar coordinates is based on the formulas:\( x = r \cos \theta \) and \( y = r \sin \theta \).When converting the rectangular equation \( x + 1 = 0 \) to polar form, we rely on these transformations. By substituting \( x = r \cos \theta \), the equation becomes \( r \cos \theta = -1 \). This is a simple way to see the positional equivalency in polar coordinates, offering a visual representation that can simplify many physical applications, especially those involving rotations and symmetry.