Problem 26 - Algebra and Trigonometry

Question

15–36 Sketch the graph of the polar equation. $$r=\cos \theta-1$$

Step-by-Step Solution

Answer

The graph of \( r = \cos \theta - 1 \) is a limaçon with an inner loop, symmetric about the x-axis.

Step 1: Understanding Polar Equations

The given polar equation is \( r = \cos \theta - 1 \). In polar coordinates, \( r \) represents the radial distance from the origin and \( \theta \) is the angle with respect to the positive x-axis.

Step 2: Determine Key Points

To sketch the graph, identify some points by calculating \( r \) for key angles \( \theta \) such as \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \).\[ r(0) = \cos(0) - 1 = 0,\] \[ r\left(\frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right) - 1 = -1,\] \[ r(\pi) = \cos(\pi) - 1 = -2,\] \[ r\left(\frac{3\pi}{2}\right) = \cos\left(\frac{3\pi}{2}\right) - 1 = -1.\]

Step 3: Analyze Symmetry and Shape

Observe that if \( r(\theta) = \cos(\theta) - 1 \), then \( r(-\theta) = \cos(-\theta) - 1 = \cos(\theta) - 1 = r(\theta) \), which indicates that the graph is symmetric about the x-axis. The nature of \( \cos \theta \) suggests that the graph will have a typical sinusoidal pattern displaced vertically.

Step 4: Sketch the Graph

Using the calculated points and symmetry, plot \( (0,0) \), \( \left(\frac{\pi}{2}, -1\right) \), \( (\pi, -2) \), and \( \left(\frac{3\pi}{2}, -1\right) \). Since \( r \) is negative for some \( \theta \), the points will be in the opposite direction of the angle, effectively reflecting across the origin.

Step 5: Draw the Complete Locus

Connect the points to form the curve. The wave-like pattern repeats every \( \pi \) units due to the periodic nature of cosine. This results in a graph resembling a shifted limaçon with an inner loop.

Key Concepts

Graphing Polar Coordinates

Polar coordinates offer a way to describe the position of a point on a plane using a distance from the origin and an angle. This is different from Cartesian coordinates, which use x and y values. The basic idea is that each point is determined by a radius, \( r \), and an angle, \( \theta \). Here are some essential concepts to grasp:

• \( r \) is the distance from the origin to the point.
• \( \theta \) is the angle measured from the positive x-axis.
• The angle \( \theta \) is measured in radians.

For instance, with the equation \( r = \cos \theta - 1 \), you determine the value of \( r \) for different angles, such as \( 0 \), \( \frac{\pi}{2} \), \( \pi \), and so on. Changes in \( \theta \) affect the distance \( r \), creating a distinct circular or looping path as you graph the points. To plot these points, remember:

• Positive \( r \): Draw the point in the direction of \( \theta \).
• Negative \( r \): Plot the point in the opposite direction of \( \theta \).

Visualizing each point is crucial in seeing the overall shape, allowing for a complete and coherent graph.

Symmetry in Polar Graphs

Symmetry can often simplify graphing polar equations. When a polar graph is symmetric, it means the graph looks the same when flipped, rotated, or moved in a certain way, and this can happen with respect to the origin or the axes. For the function \( r = \cos \theta - 1 \), symmetry is evident:

• X-axis Symmetry: If replacing \( \theta \) with \( -\theta \) yields the same function, the graph has x-axis symmetry. Here, \( r(-\theta) = \cos(-\theta) - 1 = \cos(\theta) - 1 \), indicating x-axis symmetry.
• Origin Symmetry: Points where both \( r \) and \( \theta \) change signs also suggest symmetry around the origin. This isn't directly seen in our function, but negative \( r \) values do suggest reflections.

Symmetry aids in reducing the amount of work needed to manually plot the graph. Knowing this, if you identify one part of the graph, you can predict other parts, cutting down on the points you need to calculate.

Limaçon Curves

Limaçon curves are a unique family of curves derived from equations similar to \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). These shapes are recognized by their characteristic loop or dimple, depending on the values of \( a \) and \( b \).
In the case of \( r = \cos \theta - 1 \), this is a specific type of limaçon called a "limaçon with an inner loop." Here's why:

• When \( |b| > |a| \), an inner loop forms. In the equation, \( a = -1 \) and \( b = 1 \), so \( |b| > |a| \).
• Inner loops are noticeable by the negative \( r \) values, causing the curve to intersect itself, seen at certain angles where \( r \) switches from positive to negative.
• The curve appears to shrink into the center before looping back outwards, painting the full limaçon arc.

Understanding these unique curves helps sketching because once you identify the type, you can predict key features and intersections of the curve, thus enhancing your visualization skills in polar graphs.