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Chapter 9: Problem 34

15鈥�36 Sketch the graph of the polar equation. $$r=1-2 \cos \theta \quad \text {(limacon)}$$

Short Answer

Expert verified
The limacon graph with equation \( r = 1 - 2\cos\theta \) has an inner loop and is symmetric about the x-axis.

Step by step solution

01

Understanding the Polar Equation

The given equation is \( r = 1 - 2\cos\theta \). This is a type of limacon. Limacons are characterized by their equation form \( r = a + b\cos\theta \) or \( r = a + b\sin\theta \). Here, we have \( a = 1 \) and \( b = 2 \). Since \( b > a \), we know this limacon has an inner loop.
02

Analyzing Key Values of \( \theta \)

Identify important angles \( \theta \) such as \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \) to see how \( r \) behaves:- \( \theta = 0 \): \( r = 1 - 2(1) = -1 \)- \( \theta = \frac{\pi}{2} \): \( r = 1 - 2(0) = 1 \)- \( \theta = \pi \): \( r = 1 + 2(1) = 3 \)- \( \theta = \frac{3\pi}{2} \): \( r = 1 - 2(0) = 1 \)
03

Sketching the Limacon

Using the key points identified:1. At \( \theta = 0 \), plot \( -1 \) (move along the negative x-axis since \( r \) is negative).2. At \( \theta = \frac{\pi}{2} \) and \( \frac{3\pi}{2} \), plot points at 1 unit.3. At \( \theta = \pi \), plot 3 units along the negative x-axis.Draw the curve starting and looping through these points, forming an inner loop around the pole due to the negative \( r \) at \( \theta = 0 \).
04

Verifying Symmetry and Behavior

Because the function includes \( \cos\theta \), the limacon is symmetric about the polar axis (x-axis). As \( \theta \) varies from \( 0 \) to \( 2\pi \), it covers the full shape. This symmetry confirms the graph contains an inner loop.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lima莽on Curves
A lima莽on is a special type of graph in the polar coordinate system. It appears when a polar equation has a particular form. Specifically, for a lima莽on, the equation is typically expressed as either \( r = a + b\cos\theta \) or \( r = a + b\sin\theta \). In our given equation, \( r = 1 - 2\cos\theta \), it is clear we're dealing with a lima莽on because it matches the typical form.
  • The coefficients \(a\) and \(b\) are crucial as they determine the shape of the lima莽on.
  • If \(b > a\), the lima莽on will have an inner loop, as is the case in this exercise where \(1 < 2\).
  • When \(a = b\), the lima莽on becomes a cardioid, a heart-shaped curve.
  • If \(a > b\), the curve is sometimes referred to as a dimpled lima莽on, indicating it lacks an inner loop.
Understanding these characteristics helps in predicting what the graph will look like before even plotting points.
Polar Coordinates
Polar coordinates provide a unique way to represent points on a plane. Instead of using Cartesian coordinates \((x, y)\), polar coordinates use a radius and an angle, \((r, \theta)\). This method is particularly useful in graphing equations like the lima莽on, as it naturally adapts to curves and circular paths.
  • The \(r\) value in polar coordinates tells us the distance from the origin or pole.
  • The \(\theta\) value is the angle from the positive x-axis, measured in radians.
In our exercise, for specific values of \(\theta\) such as \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\), we calculated the corresponding \(r\) values to help plot the graph effectively. This method simplifies the graphing because it aligns better with the circular shapes involved in polar equations.
Graphing Techniques for Polar Graphs
Graphing polar equations like a lima莽on involves a few essential techniques to ensure accuracy. Let's explore these strategies further.
  • Determine symmetry: Lima莽ons with cos terms are symmetric about the x-axis, which helps simplify the plotting process.
  • Identify crucial angles: Typically, angles such as \(0, \frac{\pi}{2}, \pi\), and \(\frac{3\pi}{2}\) are calculated as these provide significant points on the graph.
  • Plot key points: Use these critical angles to determine corresponding \(r\)-values, and sketch the graph by connecting these points. For example:
    • At \(\theta = 0\), \(r = -1\), indicating a point along the negative x-axis.
    • At \(\theta = \pi\), \(r = 3\), showing a point in the opposite direction.
  • Observe for loops: When \(b > a\), an inner loop is present, as \(r\) becomes negative for some \(\theta\) values.
By adopting these graphing techniques, the often challenging task of plotting polar equations becomes more straightforward and understandable, especially for intricate curves like the lima莽on.

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