Problem 47 - Calculus Early Transcendentals

Question

\(29-48\) Sketch the curve with the given polar equation. $$r=1+2 \cos 2 \theta$$

Step-by-Step Solution

Answer

The polar curve is a limacon with an inner loop and four petals, symmetrically about the polar axis.

Step 1: Analyze the Polar Equation

The given polar equation is \( r = 1 + 2 \cos(2\theta) \). This equation represents a type of polar graph called a 'rose' or 'limacon.' We notice the presence of \( \cos(2\theta) \), which implies symmetries and repetitions over the polar graph.

Step 2: Determine the Type of Graph

The equation \( r = a + b \cos(2\theta) \) where \( a = 1 \) and \( b = 2 \) indicates that this is a limacon because the form \( r = a + b \cos(n\theta) \) with \( n=2 \) suggests a rose curve with more than one petal. Since \( b > a \), it particularly has an inner loop.

Step 3: Identify Symmetry and Number of Petals

Since \( n = 2 \), our polar graph will have \( 2n = 4 \) petals. Each petal will have two symmetrical points due to the \( \cos \) component indicating symmetry about the horizontal axis (polar axis).

Step 4: Calculate Key Angles and Points

To sketch the curve, calculate \( r \) for key angles like \( \theta = 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi, \ldots \).- At \( \theta = 0 \), \( r = 1 + 2 \cos(0) = 3 \).- At \( \theta = \frac{\pi}{4} \), \( \cos(\frac{\pi}{2}) = 0 \) so \( r = 1 \).- At \( \theta = \frac{\pi}{2} \), \( \cos(\pi) = -1 \), so \( r = -1 \) (indicating the other loop).- Repeat for more angles for complete sketching.

Step 5: Sketch the Graph

Using the calculated points, begin drawing the curve in polar coordinates. Start at the petal with the longest radius which occurs at \( \theta = 0 \) and is \( r = 3 \). Connect the points smoothly, noting the symmetry and ensuring you draw both loops: one inner (negative \( r \) values) and one outer (positive \( r \) values). Complete the cycle to draw four symmetrical petals around the origin.

Key Concepts

Limacon Curve

A limacon is a specialized type of polar curve that can present various shapes, including a dimpled edge, a heart shape, or an inner loop. It's defined by equations of the form \( r = a + b \cos(n\theta) \) or \( r = a + b \sin(n\theta) \). In your given equation, \( r = 1 + 2 \cos(2\theta) \), it belongs to the limacon family. The characteristics of limacons are primarily determined by the relationship between \( a \) and \( b \):

• If \( |b| < |a| \), the limacon does not have an inner loop.


• If \( |b| = |a| \), it is known as a cardioid (heart-shaped curve).


• If \( |b| > |a| \), which is the case in the provided equation, the curve exhibits an inner loop.

Limacons can often be visually interesting and may exhibit a peculiar kind of symmetry unique to polar graphs, making them a staple for study in polar coordinate systems.

Symmetry in Polar Graphs

Symmetry plays an essential role in polar graphs, making them easier to sketch and understand. A curve's symmetry can help predict its behavior and layout, especially when determining basic features like loops or petals.For polar graphs:

• Symmetry about the polar axis means that replacing \( \theta \) with \( -\theta \) in the equation yields the same \( r \) value.


• Symmetry about the line \( \theta = \frac{\pi}{2} \) suggests replacing \( r \) with \( -r \) and still obtaining a valid point.


• The given equation \( r = 1 + 2 \cos(2\theta) \) exhibits symmetry across the polar axis due to the cosine function, which mirrors across the horizontal line through its origin.

Recognizing symmetry can enhance your graphing efficiency, allowing you to focus on a simpler range of angles to determine the full picture of the polar graph's behavior. Identifying the symmetrical aspects lets you understand those repeating patterns or directions in which the graph exhibits mirrored behavior.

Rose Curves

Rose curves are a charming type of polar graph characterized by their petal-like structures. They are defined by equations of the form \( r = a \cos(n\theta) \) or \( r = a \sin(n\theta) \). The number of petals a rose curve will exhibit depends on the angle multiplier \( n \):

• If \( n \) is even, the rose will display \( 2n \) petals.


• If \( n \) is odd, the rose will present precisely \( n \) petals.

In the equation \( r = 1 + 2 \cos(2\theta) \), the term \( 2\theta \) designates a structure resembling a rose curve with enhanced features due to the addition of a constant and amplitude term. Although primarily a limacon due to mixed features, it holds rose-like traits due to \( n = 2 \), resulting in four distinct petals. These petals create an engaging visual symmetry that repeats every \( \pi \), offering a unique insight into periodicity and symmetry in polar graphs.Understanding rose curves can be crucial for students to identify and graph these graceful curves, considering they naturally blend with other polar graph features, such as symmetry and periodic repetitions. By recognizing rose curve attributes, you can better predict and graph their captivating shapes.