Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of the given polar equation and verify its symmetry (see Examples \(1-3)\). \(r=1-2 \sin \theta\) (limaçon)

Step-by-Step Solution

Answer

The graph is a limaçon with an inner loop, symmetric about the y-axis.

Step 1: Identify the Polar Equation Type

The given polar equation is in the form of a limaçon, which is a type of polar curve. This can be recognized by the equation format, which is typically given by \( r = a + b\sin\theta \) or \( r = a + b\cos\theta \). In this case, it is \( r = 1 - 2\sin\theta \), indicating it is a limaçon.

Step 2: Determine the Type of Limaçon

To determine the type of limaçon, we compare the constants. Here, \( a = 1 \) and \( b = 2 \). Since \(|b| > |a|\), this confirms that the graph is a limaçon with an inner loop.

Step 3: Calculate Key Angles

To find key points that help sketch the graph, consider when \( r = 0 \). Solve \( 1 - 2\sin\theta = 0 \) for \( \theta \). This gives \( \sin\theta = \frac{1}{2} \), leading to \( \theta = \frac{\pi}{6} \) or \( \theta = \frac{5\pi}{6} \).

Step 4: Plot Key Points and Symmetry

For various \( \theta \) (e.g., 0, \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\)), calculate \( r \). When \( \theta = 0, r = 1 \); \( \theta = \frac{\pi}{2}, r = -1 \); \( \theta = \pi, r = 1 \); \( \theta = \frac{3\pi}{2}, r = 3 \). Symmetry can be checked: substitute \( -\theta \), \( \pi - \theta \), or \( \pi + \theta \) in the original equation. The equation is symmetric about the y-axis because substituting \( \pi - \theta \) yields the original equation.

Step 5: Sketch the Graph

Using the key points and the symmetry identified, plot the points: \((1, 0)\), point radius at \(\theta = \frac{\pi}{2}\) to \(-1\), and symmetric counterparts. The graph looks like a limaçon with an inner loop centered around the pole, intersecting negative y and extending up to \(r = 3\).

Key Concepts

Limaçon

A Limaçon is a fascinating type of polar curve that can take on various shapes depending on its parameters. It belongs to a category of curves derived from the parametric equations in physics and geometry.
An equation of a Limaçon is usually expressed as either \( r = a + b\sin\theta \) or \( r = a + b\cos\theta \). In these equations, constants \( a \) and \( b \) determine the general characteristics of the curve.

Here's what you need to know about the types of Limaçons based on these constants:

• If \( |b| \gt |a| \), the curve will have an inner loop.
• If \( |b| = |a| \), the Limaçon will be a cardioid.
• If \(|a| \gt |b| \), the curve will not have any loоps and will be dimpled or convex.

In the case of the polar equation \( r = 1 - 2\sin\theta \), we set \( a = 1 \) and \( b = -2 \). Because \(|b| \gt |a|\), we know that this Limaçon has an inner loop.
Limaçons bring diversity in shape, from loops to heart-like shapes (cardioids) and even rounded, bulgy versions. This adds an exciting layer of visual interpretation in graphing polar equations.

Symmetry in Polar Coordinates

Understanding symmetry can make plotting polar coordinates much simpler. In polar coordinates, symmetry can be checked by manipulating the angle \( \theta \) and observing how the equation responds. There are different types of symmetry to look out for:

• **Symmetry about the x-axis**: To check for this, replace \( \theta \) with \( -\theta \) and see if the equation remains unchanged.
• **Symmetry about the y-axis (or polar axis)**: Substitute \( \theta \) with \( \pi - \theta \). If the equation looks the same afterward, it is symmetric about the y-axis.
• **Symmetry about the origin**: Here, you replace \( r \) with \( -r \). If the equation doesn't change, the graph is symmetric through the origin.

For the Limaçon represented by \( r = 1 - 2\sin\theta \), substituting \( \pi - \theta \) into the equation results in the original equation. This confirms that the graph is symmetric about the y-axis, which helps in accurately sketching the plot reducing the work needed to find points only in part of the plane.

Polar Equation Sketching

Sketching polar equations can initially seem challenging, but with a structured approach, it becomes a fairly straightforward process. Begin by identifying the type of graph you're working with, like a Limaçon in this case.

Here’s how you might approach a sketch:

• **Identify the Type of Curve**: Recognize the pattern. For \( r = a + b\sin\theta \), it’s a Limaçon.
• **Calculate Key Angles**: Solve for angles where \( r = 0 \). This is crucial for marking the looping behavior.
• **Determine Important Radii**: Compute values of \( r \) for commonly used angles like \( \theta = 0, \frac{\pi}{2}, \pi, \) and \( \frac{3\pi}{2} \).
• **Check Symmetry**: Use symmetry to simplify plotting fewer points by taking advantage of symmetrical properties around y-axis.
• **Plot and Connect Points**: Once you have calculated and confirmed the symmetrical points, plot them on the polar plane and connect smoothly according to their expected shape, checking for features like loops.

Following these steps ensures the Limaçon, specifically \( r = 1 - 2\sin\theta \), is drawn accurately, showcasing its inner loop and symmetric attributes.