Problem 59 - Precalculus

Question

Graph the polar equation. \(r=-2-3 \cos \theta\)

Step-by-Step Solution

Answer

The curve is a limaçon with an inner loop because \(|b| > |a|\) (|3| > |2|).

Step 1: Understanding the Polar Equation

The given polar equation is \(r = -2 - 3 \cos \theta\). In polar coordinates, the expression involves \(r\) as a function of \(\theta\), where \(r\) represents the radius (distance from the origin), and \(\theta\) represents the angle from the positive x-axis.

Step 2: Identify the Type of Curve

The polar equation has the form \(r = a + b \cos \theta\), which is a standard form of a limaçon. Depending on the values of \(a\) and \(b\), the limaçon can have different shapes such as a cardioid, a dimpled limaçon, or a limaçon with an inner loop. Here, \(a = -2\) and \(b = -3\). Since \(|b| > |a|\), this curve will have an inner loop.

Step 3: Plot Key Points

To accurately plot the curve, we calculate \(r\) for several specific values of \(\theta\), such as \(0\), \(\frac{\pi}{3}\), \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), and \(2\pi\). For example: - \(\theta = 0\), \(r = -2 - 3 \cos(0) = -5\).- \(\theta = \pi\), \(r = -2 - 3 \cos(\pi) = 1\).- \(\theta = \frac{\pi}{2}\), \(r = -2 - 3 \cos(\frac{\pi}{2}) = -2\). These points will guide us in sketching the polar graph.

Step 4: Graph the Polar Equation

Now, plot the points on polar graph paper, noting the direction of plotting as \(\theta\) increases. The inner loop is visible for values of \(\theta\) where \(r\) is negative, meaning the graph extends in the opposite direction of \(\theta\). Continue plotting more points if needed to complete the graph smoothly, ensuring you capture the loop and the rest of the limaçon.

Key Concepts

Polar Coordinates

Polar coordinates offer an alternative way to represent points on a plane, unlike the more familiar Cartesian coordinates. In this system, a point's location is given in terms of:

• \( r \): The radial coordinate, which represents the distance from the origin.
• \( \theta \): The angular coordinate, which signifies the angle from the positive x-axis in a counterclockwise direction.

This approach is particularly useful for problems involving curves or circular paths because it simplifies the mathematics involved in working with such shapes. For example, a circle centered at the origin can easily be described in polar coordinates with a simple equation like \( r = a \), where \( a \) is the radius of the circle.
When using polar coordinates, it's crucial to understand that one value of \( (r, \theta) \) can represent multiple points in Cartesian coordinates, especially when \( r \) is negative. This occurs because a negative \( r \) implies the point is in the direction directly opposite \( \theta \). Polar coordinates allow for elegant solutions to complex geometries, especially when dealing with symmetries.

Limaçon

The term 'limaçon' describes a specific class of curves in polar coordinates. These curves have the general form:

• \( r = a + b \cos\theta \)

Based on the relative size of \( a \) and \( b \), limaçons can exhibit distinct forms:

• If \(|b| < |a|\), the limaçon is dimpled and appears somewhat heart-shaped.
• If \(|b| = |a|\), the limaçon takes the form of a cardioid.
• If \(|b| > |a|\), as is the case with the given equation \( r = -2 - 3\cos \theta \), the limaçon will exhibit an inner loop.

The inner loop occurs when the polar equation allows negative values of \( r \), indicating parts of the curve are plotted in the opposite direction of the angle \( \theta \). The limaçon features a rich variety of shapes, offering an interesting exploration in graphing polar equations.

Plotting Points

When plotting points for a polar equation, the key to accuracy is selecting strategic values of \( \theta \) to calculate \( r \). By systematically evaluating \( r \) at chosen angles, one can gather enough data points to faithfully represent the graph.
Take the example polar equation \( r = -2 - 3 \cos \theta \). For this, we compute \( r \) for angles such as:

• \( \theta = 0 \) gives \( r = -5 \).
• \( \theta = \pi \) results in \( r = 1 \).
• \( \theta = \frac{\pi}{2} \) leads to \( r = -2 \).

By connecting these points on a polar grid, one can observe how \( r \) changes as \( \theta \) varies, showcasing the cyclical and looping nature of the shape, specifically its inner loop. Additional points can fill in gaps where necessary, ensuring the entirety of the limaçon, including its loop, is accurately depicted. Plotting in polar coordinates demands attention to the direction of plotting, especially for negative \( r \), as this impacts the appearance of the graph.