Problem 34 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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:- \( \theta = 0 \): \( r = 1 - 2(1) = -1 \)- \( \theta = \frac{\pi}{2} \): \( r = 1 - 2(0) = 1 \)- \( \theta = \pi \): \( r = 1 + 2(1) = 3 \)- \( \theta = \frac{3\pi}{2} \): \( r = 1 - 2(0) = 1 \)

Step 3: 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 \( \theta = 0 \).

Step 4: 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.

Key Concepts

Limaçon Curves

A limaçon is a special type of graph in the polar coordinate system. It appears when a polar equation has a particular form. Specifically, for a limaçon, 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 limaçon because it matches the typical form.

• The coefficients \(a\) and \(b\) are crucial as they determine the shape of the limaçon.
• If \(b > a\), the limaçon will have an inner loop, as is the case in this exercise where \(1 < 2\).
• When \(a = b\), the limaçon becomes a cardioid, a heart-shaped curve.
• If \(a > b\), the curve is sometimes referred to as a dimpled limaçon, 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 limaçon, 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 limaçon involves a few essential techniques to ensure accuracy. Let's explore these strategies further.

• Determine symmetry: Limaçons 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 limaçon.