Problem 38 - Precalculus: Functions and Graphs

Question

Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\) -plane. $$r \sin \theta=-2$$

Step-by-Step Solution

Answer

The equation \(y = -2\) is the Cartesian form, representing a horizontal line at \(y = -2\). In polar coordinates, this is a continuous line at \(-2\) across all angles.

Step 1: Convert the Polar Equation to Cartesian Coordinates

The given polar equation is \(r \sin \theta = -2\). We know the relationship between polar and Cartesian coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). Therefore, \(y = r \sin \theta\). By substituting this into the polar equation, we get \(y = -2\).

Step 2: Describe the Graph of the Cartesian Equation

The Cartesian equation \(y = -2\) represents a horizontal line that crosses the y-axis at -2. This line is parallel to the x-axis and does not vary with respect to \(x\).

Step 3: Sketch the Graph in the Polar Plane

In the polar plane, the equation \(r \sin \theta = -2\) translates to a set of points where the radial distance, \(r\), places the point on the line \(y = -2\) in Cartesian coordinates. Since \(r\) must be negative in this equation to reach \(-2\) at some angle \(\theta\), the line is essentially wrapping around at all angles. This appears as a continuous line parallel to the \(\theta\)-axis, effectively creating a band around \(-2\) from all directions.

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using two parameters: the radial distance and the angle. Unlike the Cartesian system, which uses horizontal and vertical distances, polar coordinates locate a point by its distance from the origin (denoted as \(r\)) and the angle (\(\theta\)) formed with the positive x-axis.
Understanding polar coordinates involves:

• Radial Distance (\(r\)): This tells you how far away the point is from the origin. If \(r\) is positive, the point is in the direction of \(\theta\); if negative, it is in the opposite direction.
• Angle (\(\theta\)): Measured in radians or degrees, \(\theta\) tells you the direction from the positive x-axis to the point.

This system is especially useful in scenarios where relationships can be better expressed through angles and radial dimensions, like in circular or spiral patterns.

Cartesian Coordinates

The Cartesian coordinate system represents points through two values: x and y coordinates. This method forms the basis for much of two-dimensional graphing.
In this system, any point is defined by:

• X-coordinate: The horizontal distance of the point from the y-axis.
• Y-coordinate: The vertical distance of the point from the x-axis.

These coordinates are used to describe locations in rectangular grids, which makes plotting mathematical functions and shapes straightforward. Transforming polar coordinates into Cartesian ones allows for easier interpretation and graphing using familiar horizontal and vertical lines.

Horizontal Line Equation

A horizontal line in the Cartesian coordinate system is simple, as it remains constant regardless of the x-value. The general equation of a horizontal line is \(y = b\), where \(b\) is a fixed value indicating where the line crosses the y-axis.
Key aspects of horizontal lines include:

• They never intersect the x-axis unless \(b = 0\).
• All points on the line have the same y-coordinate.
• The line is parallel to the x-axis indicating a slope of zero.

For example, the equation \(y = -2\) represents a horizontal line 2 units below the x-axis, unaffected by any changes in the x-coordinate.

Coordinate Transformation

Coordinate transformation involves converting between different coordinate systems, such as from polar to Cartesian coordinates. This conversion lets us use the characteristics of different systems in varied applications.
The main formulas for transforming polar to Cartesian coordinates are:

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

Using these equations, the polar equation \(r \sin \theta = -2\) becomes \(y = -2\) in Cartesian form. This step is crucial because it allows us to visualize the graph in a standard rectangular grid, making interpretation and analysis more intuitive.
When converting coordinates, it’s important to understand the context and characteristics of each coordinate system to accurately represent and interpret the data.