Problem 25 - Precalculus Mathematics for Calculus

Question

Sketch a graph of the polar equation. $$r=2-2 \cos \theta$$

Step-by-Step Solution

Answer

The graph is a cardioid symmetric to the polar axis with a loop touching the origin.

Step 1: Identify the Type of Polar Equation

The given equation is in the form \( r = a - b \cos \theta \). This is a type of limaçon (specifically a cardioid when \( a = b \)). For \( r = 2 - 2\cos\theta \), \( a \) and \( b \) are both equal to 2, so the graph is a cardioid.

Step 2: Determine Symmetry

Analyze the equation to determine symmetry. Since it contains \( \cos \theta \), it is symmetric with respect to the polar axis (the horizontal axis).

Step 3: Find Key Points

Calculate for key angles \( \theta \) to find corresponding \( r \). Start with \( \theta = 0\), \( \pi/2\), \( \pi\), \(3\pi/2\), and \( 2\pi \). At \( \theta = 0 \), \( r = 2 - 2\cdot 1 = 0 \). At \( \theta = \pi/2 \), \( r = 2 \). At \( \theta = \pi \), \( r = 2 + 2\cdot 1 = 4 \). At \( \theta = 3\pi/2 \), \( r = 2 \). At \( \theta = 2\pi \), \( r = 0 \).

Step 4: Sketch the Graph

Using the key points, we can now sketch the graph of the equation. Start at the pole (\( r = 0, \theta = 0 \)) and trace through \( r = 2, \theta = \pi/2 \), \( r = 4, \theta = \pi \), and \( r = 2, \theta = 3\pi/2 \), returning to \( r = 0, \theta = 2\pi \). The result is a heart-shaped graph touching the pole at \( \theta = 0 \) and symmetric about the polar axis.

Key Concepts

Limaçon

The limaçon is a fascinating type of curve that appears in polar coordinate systems. Limaçons are defined by equations of the form \( r = a \, \pm \, b \cos \theta \) or \( r = a \, \pm \, b \sin \theta \). They are known for their diverse shapes, which can transform from having an inner loop to a dimpled form depending on the values of \( a \) and \( b \).
When \( a = b \), the limaçon forms a special kind called a cardioid. If \( a < b \), the limaçon will have an inner loop, creating a shape that looks like a distorted circle. When \( a > b \), the shape is limaçon dimpled but does not self-intersect.
To visualize limaçon effectively, plot key points at critical angles such as \( \theta = 0, \pi/2, \pi, 3\pi/2, \) and \( 2\pi \), observing changes in radial distance \( r \). These variations highlight how the limaçon evolves across different angles, offering insight into its structure.

Cardioid

A cardioid is a distinctive heart-shaped curve resulting from a special case of the limaçon. It emerges when the parameters \( a \) and \( b \) in the equation \( r = a - b \cos \theta \) are equal, meaning \( a = b \).
Cardioids possess a unique symmetry and they are visually appealing, due to their smooth and continuous curve that starts and ends at the pole. They are symmetrical about the polar axis when the \( \cos \theta \) function is used.
For example, the cardioid given by the equation \( r = 2 - 2 \cos \theta \) consists of key points where the shape bends and curves:

• At \( \theta = 0 \), \( r = 0 \), causing the cardioid to touch the pole.
• At \( \theta = \pi/2 \), \( r = 2 \), creating a bulge in the graph.
• At \( \theta = \pi \), \( r = 4 \), placing the farthest point from the pole.

The cardioid's structure is elegant, defined by seamless transitions across all angles.

Symmetry in Polar Graphs

Symmetry plays a crucial role in plotting and understanding polar graphs. It helps in predicting curve behavior and simplifying graph construction.
There are three primary symmetry types in polar graphs:

• Symmetry about the polar axis: This is evident when the polar equation contains \( \cos \theta \). For example, in equations like \( r = 2 - 2 \cos \theta \), the graph is symmetric with respect to the horizontal axis, meaning the shape mirrors on both sides of the axis.
• Symmetry about the line \( \theta = \frac{\pi}{2} \): Present in graphs involving \( \sin \theta \), where the curve is mirrored across the vertical axis.
• Symmetry about the pole: The entire graph can map onto itself if \( r(\theta) = -r(\theta + \pi) \).

Recognizing these symmetries allows one to sketch only part of the graph and replicate it through symmetry rules, streamlining the graphing process.