Problem 27 - Thomas Calculus

Question

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r=4 \csc \theta $$

Step-by-Step Solution

Answer

The graph is a horizontal line at \( y = 4 \).

Step 1: Understanding the Polar Equation

The given equation is \( r = 4 \csc \theta \). In polar coordinates, \( r \) is the radius or distance from the origin and \( \theta \) is the angle. The cosecant function, \( \csc \theta \), is the reciprocal of the sine function, so \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Substitute \( \csc \theta \) with .

Using the identity \( \csc \theta = \frac{1}{\sin \theta} \), we can rewrite the equation as \( r = \frac{4}{\sin \theta} \).

Step 3: Multiply Both Sides by \( \sin \theta \)

Multiply both sides of the equation by \( \sin \theta \) to isolate \( r\sin\theta \) on the left side: \( r\sin\theta = 4 \).

Step 4: Convert to Cartesian Coordinates

Recall that in polar coordinates, \( r\sin\theta = y \). Therefore, the equation \( r\sin\theta = 4 \) can be rewritten in terms of Cartesian coordinates as \( y = 4 \).

Step 5: Identify the Graph

The equation \( y = 4 \) describes a horizontal line in the Cartesian coordinate system at a distance 4 units above the x-axis.

Key Concepts

Polar equations

Polar equations are mathematical expressions that relate the radius and angle of a point in the polar coordinate system. In this system, each point is defined by a pair \(r, \theta\), where \(r\) is the distance from the origin (or pole), and \(\theta\) is the angle from the positive x-axis, measured in a counterclockwise direction. This system is useful in scenarios where the relationship between variables naturally revolves around a central point.
For instance, the equation \( r = 4 \csc \theta\) is a polar equation. Such equations often describe circles, spirals, and other shapes that are best represented with a center-based approach.
Converting polar equations to Cartesian equations involves using relationships between polar and Cartesian coordinates. This makes the visualization of graphs easier in the more familiar Cartesian system.

Cartesian coordinates

The Cartesian coordinate system is a two-dimensional system where each point is defined by an ordered pair, denoted as \(x, y\). This system is named after René Descartes and is the most commonly used coordinate system in mathematics.
In this system:

• \(x\) represents the horizontal position of a point relative to the origin.
• \(y\) indicates the vertical position of a point relative to the origin.

To convert from polar to Cartesian coordinates, we use the following relationships:

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

These formulas allow us to express points in terms of their horizontal and vertical components, making it easier to graph functions such as the line \(y = 4\) derived from our polar equation.

Cosecant function

The cosecant function is a trigonometric function, denoted \(\csc \theta\), and is defined as the reciprocal of the sine function. Thus, \[ \csc \theta = \frac{1}{\sin \theta} \] This function is undefined at angles where the sine value is zero, specifically at integer multiples of \(\pi\) (0, \(\pi\), 2\(\pi\), etc.).
In the context of polar equations, the cosecant function often arises as components of curves where radial distances vary inversely with the sine of the angle.
By recognizing that \( r = 4 \csc \theta \) translates into \( r = \frac{4}{\sin \theta}\), we can manipulate the equation to explore its characteristics in Cartesian format, thus assisting in graph interpretation.

Graph identification

Graph identification involves determining what type of geometric figure is represented by an equation. This is an essential skill for visualizing mathematical relationships.
After converting our polar equation \(r = 4 \csc \theta\) into the Cartesian equation \(y = 4\), the graph can be clearly identified.

• A horizontal line at \(y = 4\) indicates a straight line parallel to the x-axis, positioned 4 units above it.

This ability to identify and visualize graphs aids significantly in understanding geometric relationships, allowing us to deduce properties about the line, like its slope and intercepts, and how it interacts with other mathematical elements. This understanding connects abstract concepts with visual representations, fostering a deeper insight into mathematics.