Problem 25 - Thomas Calculus

Question

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r \sin \theta=0 $$

Step-by-Step Solution

Answer

The graph is a horizontal line, represented by \( y = 0 \).

Step 1: Recall Polar to Cartesian Conversion

In polar coordinates, every point is defined by a radius \( r \) and an angle \( \theta \). To convert this to Cartesian coordinates (\( x, y \)), use the following relationships: \( x = r \cos \theta \), and \( y = r \sin \theta \).

Step 2: Substitute Polar Equation

Given the polar equation \( r \sin \theta = 0 \), substitute the Cartesian conversion \( y = r \sin \theta \). The equation becomes \( y = 0 \).

Step 3: Identify the Graph

The equation \( y = 0 \) represents a horizontal line along the x-axis on a Cartesian plane. Every point on this line has a y-coordinate of 0.

Key Concepts

Cartesian Coordinates

Cartesian coordinates are a way of determining the position of a point on a plane using two numbers. These numbers are known as the coordinates and are typically denoted as \(x, y\). These coordinates specify how far the point is from the x-axis and y-axis.

The x-coordinate (abscissa) represents the horizontal position, while the y-coordinate (ordinate) indicates the vertical position. By using these two values, any point in a two-dimensional space can be precisely located.

When converting from polar to cartesian coordinates, the relationships are crucial:

• \(x = r \cos \theta\), where \(r\) is the radius and \(\theta\) is the angle in radians.
• \(y = r \sin \theta\), linking the y-coordinate directly to the polar definition.

These formulas allow us to transform polar coordinates into the familiar Cartesian plane, revealing how points relate to each other in standard x-y terms.

Polar Equations

Polar equations describe relationships between points in a polar coordinate system, where each point is defined by a radius \(r\) and angle \(\theta\). This differs from Cartesian coordinates as it relies on the position relative to the origin, rather than the axes.

The given polar equation \(r \sin \theta = 0\) is an example of a simple relationship where the value of \(y\) is zero. When substituting into the Cartesian system, the equation becomes \(y = 0\).

This transformation shows how polar equations can directly influence and define lines or curves within the Cartesian system. Certain equations in polar form make use of sine, cosine, and even tangent to describe complex shapes, once converted, they often show familiar forms such as circles, ellipses, or, in this case, lines.

Graph Identification

Graph identification involves recognizing the visual representation of mathematical equations on a plane, such as the Cartesian plane. Understanding this concept is essential for visualizing relationships described by equations.

In the original problem, the polar equation transformed into \(y = 0\). This resulting Cartesian equation indicates a horizontal line along the x-axis, as each point on this line has \(y = 0\).

The ability to identify this graph from its equation helps in:

• Visualizing mathematical concepts.
• Understanding relationships between variables.
• Simplifying complex problems into understandable parts.

Graph identification ensures that abstract equations translate into concrete, understandable visuals, critical for both basic and advanced mathematics studies.