Problem 39 - 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. $$r=-3 \csc \theta$$

Step-by-Step Solution

Answer

The graph is a horizontal line at \(y = -3\) in Cartesian coordinates.

Step 1: Understanding the Polar Equation

The given polar equation is \(r = -3 \csc \theta\). Recall that \(\csc \theta\) is the reciprocal of \(\sin \theta\). Thus, the equation can be rewritten as \(r = \frac{-3}{\sin \theta}\).

Step 2: Conversion from Polar to Cartesian Coordinates

To convert from polar to Cartesian coordinates, we use the equations \(x = r \cos \theta\) and \(y = r \sin \theta\). Since \(r = \frac{-3}{\sin \theta}\), we substitute in the equation for \(y\): \(y = \left(\frac{-3}{\sin \theta}\right)\sin \theta = -3\).

Step 3: Deriving the Cartesian Equation

From the conversion, we have found \(y = -3\). This is the equation of a horizontal line in the Cartesian coordinate system, meaning its graph remains constant for all \(x\)-values.

Step 4: Identify the Graph in the Polar Plane

In the polar coordinate system, the equation \(r = -3 \csc \theta\) indicates the radius \(r\) is not dependent on the angle \(\theta\) as long as \(\theta\) is such that \(\sin \theta eq 0\). This corresponds to any point where \(y = -3\) on the graph.

Step 5: Graph Sketching in Polar and Cartesian Coordinates

Thus, the graph of the equation in Cartesian coordinates is a horizontal line at \(y = -3\). In the \(r\) vs. \(\theta\) plane, this corresponds to all angles \(\theta\), except where \(\sin \theta = 0\), with a radius \(r = -3/\sin \theta\), indicating the line at \(y = -3\).

Key Concepts

Polar coordinates

Polar coordinates are a way to describe the location of a point using a combination of distance and direction. In this system, each point is determined by a distance from a reference point (often the origin) and an angle from a reference direction.
They are especially useful in scenarios where the relationship between points is more naturally described in terms of angles and distances, such as problems involving circular or rotational symmetry.In a polar coordinate system:

• The 'distance' from the origin is denoted as \(r\).
• The 'angle' is indicated as \(\theta\), usually measured in radians or degrees from a fixed direction.

For example, the polar equation \(r = -3 \csc \theta\) connects the radius directly with the angle \(\theta\). In our particular case, the personality of the curve is defined significantly by the relationship of radius \(r\) with the angle direction \(\theta\). This allows us to explore the graph behavior for a range of angles.

Cartesian coordinates

Unlike polar coordinates, Cartesian coordinates describe a point in a plane by specifying its horizontal and vertical distances from two mutually perpendicular lines often called axes. This system uses two numbers, \(x\) and \(y\).

• The \(x\)-coordinate represents the horizontal position relative to the origin.
• The \(y\)-coordinate represents the vertical position relative to the origin.

In the example problem, the polar equation \(r = -3 \csc \theta\) was transformed into a horizontal line at \(y = -3\) in the Cartesian coordinate plane. This transformation illustrates how a curve defined by an angular equation in polar coordinates can become a simple line when described using Cartesian coordinates.

Coordinate conversion

The process of converting between polar and Cartesian coordinates involves a set of mathematical relationships between \(r, \theta\) and \(x, y\). This conversion is crucial for problems where you need to understand or modify data represented in one system to another.To convert polar coordinates \((r, \theta)\) to Cartesian \((x, y)\), the following formulas are used:

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

For the exercise, when \(r = \frac{-3}{\sin \theta}\), substituting into the equation for \(y\) gives \(y = -3\). This demonstration of conversion is a key skill, helping students visualize how different types of graphs can relate across these two coordinate systems.

Graphing equations

Graphing equations is an essential skill for visualizing mathematical relationships. It helps in understanding how equations transform across different systems, such as from polar to Cartesian coordinates.For the original exercise, the polar equation \(r = -3 \csc \theta\) leads to graphing the line \(y = -3\) in a Cartesian plane. Here are steps involved in graphing such equations:

• Identify the type of equation you are dealing with, whether it's polar or Cartesian.
• Convert the equation, if needed, using the appropriate conversion methods.
• Graph the resulting equation to visualize it. The equation \(y = -3\) is straightforward in that it creates a horizontal line across the graph.

Graphing helps confirm that our conversion between polar and Cartesian coordinates is accurate, as a mismatch would indicate errors in the conceptual understanding or mathematical steps. This process enriches the intuition on how data can be interpreted or visualized using either coordinate system.