Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=\frac{3}{\sin \theta} $$

Step-by-Step Solution

Answer

The curve is a line at \( y = 3 \) with no eccentricity, as it is a degenerate conic.

Step 1: Convert to Cartesian Form

Start by multiplying both sides by \( \sin \theta \) to clear the fraction: \[ r \sin \theta = 3. \] Using the polar to Cartesian conversion \( r \sin \theta = y \), substitute to get \( y = 3 \). This is the Cartesian equation of a line parallel to the x-axis, three units above it.

Step 2: Identify the Curve

The equation \( y = 3 \) represents a straight line in the Cartesian coordinate system. However, in polar coordinates, \( r = \frac{3}{\sin \theta} \) is a special case that describes a conic section, specifically a "vertical" line, represented in this system as a degenerate conic.

Step 3: Determine Eccentricity

Since the problem involves the form \( r = \frac{ed}{1 - e \sin \theta} \), compare with \( r = \frac{d}{\sin \theta} \) to see that \( e = 0 \). An eccentricity of zero indicates a circle, but since this degenerates to a line in polar form, it lacks actual eccentricity.

Step 4: Sketch the Graph

Graph \( r = \frac{3}{\sin \theta} \) as a horizontal line at \( y = 3 \) on the Cartesian plane. In polar coordinates, it's represented as points where the distance from the pole (origin) is undefined or goes to infinity along \( \theta = 0 \) and reflected across it, but appears as a straight vertical trajectory or asymptote line.

Key Concepts

Cartesian Conversion

In polar coordinates, it's often useful to convert equations into Cartesian form to better understand their geometric characteristics. When we have an equation like \( r = \frac{3}{\sin \theta} \), our first step is to clear the fraction by multiplying both sides by \( \sin \theta \), giving us \( r \sin \theta = 3 \).

Then, using the conversion formula from polar to Cartesian coordinates where \( r \sin \theta = y \), we substitute and simplify to get the equation \( y = 3 \). This equation represents a horizontal line three units above the x-axis.

The process of Cartesian conversion often reveals the underlying simplicity of a curve that might appear complex in polar form. It's a key skill in mathematics for transforming and understanding geometric shapes in different coordinate systems.

Conic Sections

Conic sections are the curves that can be created as intersections of a cone with a plane. These include circles, ellipses, parabolas, and hyperbolas.

In our exercise, though, the equation \( r = \frac{3}{\sin \theta} \) does not represent a typical conic. While in polar coordinates it can seem like a conic section due to its structure, its transformation to a Cartesian plane results in the equation \( y = 3 \), which is simply a line.

• Lines in polar coordinates can sometimes be seen as a degenerate conic section, as they lack the curved nature typical of other conics.
• In this specific scenario, the polar form of a line aligns with the concept of a conic section due to its representation in polar equations as vertical or horizontal.

Understanding how and when lines appear as conics is crucial for grasping more complex polar equations and further mathematical ideas.

Eccentricity

Eccentricity is a measure of how much a conic section deviates from being circular. However, in the case of our example \( r = \frac{3}{\sin \theta} \), the notion of eccentricity becomes a bit unconventional.

For most conics:

• A circle has an eccentricity of 0.
• An ellipse has an eccentricity between 0 and 1.
• A parabola has an eccentricity of 1.
• A hyperbola has an eccentricity greater than 1.

In this exercise, though, since the equation results in a line in both polar and Cartesian forms, traditional measures of eccentricity do not apply. When expressed in conic form, it may seem the line transforms to a circle with eccentricity \( e = 0 \), but practically, it is just a line bearing no actual eccentricity. This highlights an important concept: not all equations that look like conics truly embody conic characteristics when interpreted geometrically.

Graph Sketching

Graph sketching is about visualizing data or functions to better understand their structure and characteristics. For the equation \( r = \frac{3}{\sin \theta} \), the conversion to \( y = 3 \) in Cartesian coordinates simplifies the sketching process.

In the Cartesian plane:

• Simply draw a horizontal line parallel to the x-axis, exactly three units above it.
• This line extends infinitely in both directions, consistent with the nature of lines.

In polar coordinates:

• The line appears as a series of points where the radius can become infinitely large as \( \theta \) approaches zero, creating a visual of a vertical path that's undefined.

This exercise helps in understanding how lines manifest differently depending on the coordinate system used. It also illustrates the importance of mastering how to sketch and interpret graphs in both polar and Cartesian formats.