Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 23-36, name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. \(r=4 \sin \theta\)

Step-by-Step Solution

Answer

The curve is a circle with radius 2 centered above the origin; its eccentricity is 0.

Step 1: Identify the type of curve

The given polar equation is of the form \( r = a \, \sin \theta \). These equations represent conic sections. Specifically, equations of this form describe a circle when \( a \) equals any constant.

Step 2: Determine the parameters

In the given equation \( r = 4 \sin \theta \), we can see that \( a = 4 \). This means we have a circle. This circle is centered at the pole (origin) since it is a type of rose curve when the value of \( n = 1 \), which is not explicitly used here.

Step 3: Confirm eccentricity

For the polar equation \( r = a \sin \theta \), eccentricity \( e \) is used to determine the type of conic. Since this equation represents a circle, the eccentricity is \( e = 0 \).

Step 4: Sketch the graph

To sketch the graph of \( r = 4 \sin \theta \), you start by noting that the maximum value of \( r \) occurs when \( \sin \theta = 1 \), which gives \( r = 4 \). As \( \theta \) changes from 0 to \( \pi \), the value of \( \sin \theta \) increases from 0 to 1 and then back to 0, so the circle is above the polar axis (the horizontal axis) with a radius of 2 centered above the pole.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. These intersections can result in different types of curves, namely:

• Circles
• Ellipses
• Parabolas
• Hyperbolas

In polar coordinates, these conic sections have unique representations. For instance, the polar equation \( r = a \sin \theta \) typically describes a conic section where the resulting shape and position depend on the value of \( a \) and the function (sine or cosine) used. In this context, a circle is a special kind of conic where the eccentricity is zero. In simpler terms, it is a perfectly round shape centered at the pole, with all points equidistant from a central point.

Eccentricity

Eccentricity is a crucial concept in understanding conic sections as it defines the shape of the conic. It is represented by \( e \), and different values of eccentricity correspond to different conics:

• \( e = 0 \): Circle
• \( 0 < e < 1 \): Ellipse
• \( e = 1 \): Parabola
• \( e > 1 \): Hyperbola

For the polar equation \( r = 4 \sin \theta \), where the result is a circle, the eccentricity \( e \) is \( 0 \). This means, in the given problem, the curve is uniformly spaced around the center. No single point stretches further away from the center compared to others, resulting in a symmetric round shape.

Graph Sketching

Sketching graphs of polar equations involves plotting points based on the polar coordinate's angle \( \theta \) and radius \( r \). For the equation \( r = 4 \sin \theta \), this process is both interesting and intuitive. Here's how we can break it down:

• When \( \theta = 0 \), \( \sin \theta = 0 \), so \( r = 0 \). The point starts at the pole.
• At \( \theta = \frac{\pi}{2} \), \( \sin \theta = 1 \), giving the maximum radius of 4.
• As \( \theta \) increases from 0 to \( \pi \), \( r \) grows to 4 and falls back to 0.

This indicates the circle is stretched across the polar axis. During graph sketching, you consider these points and connect them smoothly, forming the complete circle. It visually verifies that the curve extends perfectly above the horizontal polar axis and returns to the origin, emphasizing the circle's centered and equidistant form.