Problem 55 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 45-78: Sketch the graph of the polar equation. $$ r=-6(1+\cos \theta) $$

Step-by-Step Solution

Answer

The graph of this polar equation is a limaçon with an inner loop.

Step 1: Recognize the Type of Polar Equation

The given polar equation is \( r = -6(1 + \cos \theta) \), which is a form of the limaçon family. Specifically, this is in the form \( r = a + b \cos \theta \), where \( a = -6 \) and \( b = -6 \). Since \( a = b \), we know this is a limaçon with an inner loop.

Step 2: Convert to Rectangular Coordinates for Plotting Points

Using the identities \( x = r \cos \theta \) and \( y = r \sin \theta \), you can express the equation in rectangular coordinates. This isn't necessary for sketching but helpful to understand the shape in Cartesian terms.

Step 3: Identify Key Points and Symmetries

Since the equation is symmetric concerning the polar axis (positive x-axis), we can use this symmetry to simplify plotting. Some key points to plot could include \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \), etc. For example, when \( \theta = 0 \), we have \( r = -12 \), indicating that the curve crosses the polar axis.

Step 4: Determine the Behavior at Specific Angles

Calculate \( r \) for several \( \theta \) values to determine significant behavior. For example, when \( \theta = \pi \), \( r = 0 \), reaching the origin. At \( \theta = \frac{\pi}{3} \) or any angle resulting in \( -6(1 + \frac{1}{2}) \), \( r = -9 \), indicating the inner loop.

Step 5: Sketch the Polar Graph

Using the key points and the symmetry properties, begin by plotting the inner loop, which occurs between \( \theta = 0 \) and \( \theta = \pi \). Then plot the outer boundary of the limaçon which exists as \( r \) follows the equation after crossing the pole. Ensure the loop reflects the negative value of \( r \) and correct directional plotting.

Key Concepts

Limaçon

The limaçon is a fascinating type of mathematical curve that belongs to the broader family of polar equations. It is classified under the form \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). These curves take diverse shapes based on the relationship between the values of \( a \) and \( b \). In our given example, \( r = -6(1 + \cos \theta) \), both \( a \) and \( b \) are equal to -6, which results in a limaçon with an inner loop. This form arises when \( a = b \), providing a distinct looping feature. The appearance of a limaçon can vary significantly:

• If \( |a| < |b| \), an inner loop is formed.
• If \( |a| = |b| \), it creates a limaçon with a cusp.
• If \( |a| > |b| \), the limaçon appears as a dimpled curve.

Using these guides, you can identify the nature of the curve simply by inspecting the equation. This makes plotting curves in polar coordinates more intuitive.

Polar Coordinates

Polar coordinates are a way of defining a point on a plane using a distance and an angle relative to a reference point and direction, usually the origin and the positive x-axis, respectively. While most are familiar with Cartesian coordinates \((x, y)\), polar coordinates use \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle measured counterclockwise from the positive x-axis. Polar coordinates are especially useful for representing curves and points that exhibit radial symmetry, such as circles and spirals. To convert between polar and Cartesian coordinates, the following equations are used:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

Conversely, given \(x\) and \(y\), you find \(r\) and \(\theta\) by:

• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)

Understanding polar coordinates opens up new ways to visualize and graph mathematical functions, making complex graphs simpler to analyze.

Symmetry in Polar Graphs

Symmetry in polar graphs makes the graphing process more efficient and manageable by allowing the symmetry properties to simplify computation or estimation of the curve's full geometry. There are several types of symmetries to consider when sketching polar graphs:

• **Symmetry with Respect to the Polar Axis:** If replacing \(\theta\) with \(-\theta\) gives an equivalent equation, the graph is symmetric with respect to the polar axis (x-axis).
• **Symmetry with Respect to the Line \(\theta = \frac{\pi}{2}\):** If replacing \(r\) with \(-r\) or \(\theta\) with \(\pi - \theta\) gives an equivalent equation, there is symmetry about the line \(\theta = \frac{\pi}{2}\) (y-axis).
• **Symmetry with Respect to the Pole:** If substituting \(r\) for \(-r\) gives an equivalent equation, then the graph has rotational symmetry through the pole (origin).

In the example equation \(r = -6(1 + \cos \theta)\), there's symmetry concerning the polar axis. This means part of the graph can be plotted and the rest can be inferred, making sketching quicker and less prone to error. Recognizing these symmetries not only helps when drawing graphs by hand but can also aid in verifying results when using graphing software.