Problem 29 - Thomas Calculus

Question

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r \sin \theta=0$$

Step-by-Step Solution

Answer

The Cartesian equivalent is \( y = 0 \), the x-axis, a horizontal line.

Step 1: Understand Polar Equation

The given polar equation is \( r \sin \theta = 0 \). In polar coordinates, this represents a condition where the \( y \)-coordinate in Cartesian coordinates is zero.

Step 2: Convert to Cartesian Coordinates

To convert from polar to Cartesian coordinates, use the relations \( x = r \cos \theta \) and \( y = r \sin \theta \). Since \( r \sin \theta = y \), the given equation \( r \sin \theta = 0 \) means \( y = 0 \).

Step 3: Identify the Graph

The Cartesian equation \( y = 0 \) is equivalent to the \( x \)-axis in the Cartesian plane, which is a straight line along the horizontal direction.

Key Concepts

Polar Coordinates

Polar coordinates offer an alternative method to describe points in a plane using a radius and an angle. Instead of referencing a fixed point along an x-axis or y-axis, points are determined from a central point or origin. The main components are:

• Radius (r): This is the distance from the origin to the point.
• Angle (\(\theta\)): Measured in radians or degrees, it determines the direction of the point relative to the positive x-axis.

To understand how these coordinates work, imagine drawing a straight line from the origin to a point. The angle \(\theta\) defines how far you rotate from the x-axis to get to this line, and the radius is simply how long the line is. This system is particularly useful in scenarios where direction and distance from a central point outweigh absolute position.

Cartesian Coordinates

In contrast to polar coordinates, Cartesian coordinates use a grid-based system. With two perpendicular axes—the x-axis (horizontal) and the y-axis (vertical)—every point in this coordinate system is defined by a pair of numbers \((x, y)\). This system is straightforward:

• x-coordinate: Indicates how far to the left or right the point is from the origin.
• y-coordinate: Indicates how far up or down the point is from the origin.

To convert from polar to Cartesian coordinates, apply the equations:\[x = r \cos \theta\]\[y = r \sin \theta\]These relationships allow for transformation between the two systems. For example, from our problem, we learned that when \(r \sin \theta = 0\), it translates to \(y = 0\), describing a horizontal line at the origin.

Graph Identification

Identifying graphs through polar and Cartesian equations involves understanding the geometric representation of algebraic conditions. In the given exercise, we convert the polar equation \(r \sin \theta = 0\) to a Cartesian form \(y = 0\). This step helps in translating the condition into a familiar visual graph form.

• For polar equations, the graph often rotates or spirals, depending on how \(r\) and \(\theta\) vary.
• For Cartesian equations, a simplified model such as straight lines, parabolas, and circles can often be drawn.

In this case, after the conversion, the graph of \(y = 0\) is a straightforward representation of the x-axis. It is a simple horizontal line running across the Cartesian plane where every point along this line has a zero y-coordinate. In practical terms, this means it intersects the location where polar conditions equate to having no vertical displacement from the origin.