Problem 22 - Thomas Calculus

Question

Graph the limacons. Limacon ("lee-ma-sahn") is Old French for "snail." You will understand the name when you graph the limacons in Exercise 21. Equations for limacons have the form \(r=a \pm b \cos \theta\) or \(r=a \pm b \sin \theta .\) There are four basic shapes. Cardioids a. \(r=1-\cos \theta \quad\) b. \(r=-1+\sin \theta\)

Step-by-Step Solution

Answer

Both equations form cardioids, with (a) symmetric about the x-axis and (b) about the y-axis.

Step 1: Identify the Equation Type

The given equations are of the form \(r = a + b \cos \theta\) or \(r = a + b \sin \theta\), these are known as limacons. Limacons have different forms depending on the values of \(a\) and \(b\). In this case, both equations are in the form of cardioids where \(a = b\), causing the limacon to take the shape of a heart.

Step 2: Analyze Equation (a)

For equation (a), \(r = 1 - \cos \theta\), identify that it is a cardioid because \(a = 1\) and \(b = 1\). The equation is symmetric about the x-axis, as it involves \(\cos \theta\). This cardioid will have its cusp oriented towards the left at the origin (0,0).

Step 3: Analyze Equation (b)

For equation (b), \(r = -1 + \sin \theta\), similarly, \(a = -1\) and \(b = 1\). This also forms a cardioid because the absolute values of \(a\) and \(b\) are equal. Since it involves \(\sin \theta\), the graph will be symmetric about the y-axis, and the cusp will point downward.

Step 4: Graph the Equations

To graph these, start by plotting points for \(\theta\) across a range of values, typically from \(0\) to \(2\pi\), and calculate \(r\) using the given equations. Connect the points to form the closed loop characteristic of these cardioids. For both equations, make sure to note the direction of the cusp based on the identified symmetry in Steps 2 and 3.

Step 5: Validate the Graphs

Check the graphs against the expected shapes. Both equations should form loops with a heart-like appearance, confirming both are cardioids. Ensure the symmetry and cusp positions align with the aforementioned analysis of each equation.

Key Concepts

Cardioid

A cardioid is a special type of limacon that takes on a heart-like shape. These curves are unique since they are characterized by the algebraic condition where the coefficients of their polar equations are equal; specifically, the equation takes the form \(r = a \pm a \cos \theta\) or \(r = a \pm a \sin \theta\). Cardioids are fascinating curves due to their distinct appearance and properties. They possess a single cusp, a point at which the curve changes direction sharply.

The name 'cardioid' is derived from the Greek word "kardia," meaning heart, fitting its shape. In polar curves, cardioids are a classic example of how mathematical equations can represent recognizable shapes. The two equations provided, \(r = 1 - \cos \theta\) and \(r = -1 + \sin \theta\), are both cardioids, as the absolute value of their coefficients (\(a = b\)) is the same, leading to that familiar heart shape in the polar plot.

Polar Coordinates

Polar coordinates are an alternative system to Cartesian coordinates for describing the locations of points in a plane. Instead of using x and y coordinates, polar coordinates use a distance from the origin, represented by \(r\), and an angle from the positive x-axis, denoted by \(\theta\). This system is particularly useful in graphing equations that have circular or spiral patterns, like cardioids and other limacons.

When using polar coordinates, any point can be expressed as \((r, \theta)\), where \(r\) is the radius or length from the origin, and \(\theta\) is the angle from the polar axis. This approach is ideal for expressing complex curves that might not easily fit into the typical grid-like pattern of Cartesian coordinates. Limacons take full advantage of the polar coordinate system, as their radial symmetry and form demand a representation that easily adapts to round shapes.

Graphing Techniques

Graphing in polar coordinates requires understanding how changes in \(\theta\) affect \(r\), the radius. Unlike Cartesian graphs, where the x and y axes stand perpendicular, polar graphs use concentric circles radiating from a point. To graph equations like a cardioid, start by creating a table of values for \(\theta\) ranging from \(0\) to \(2\pi\). Each \(\theta\) value corresponds to a specific \(r\), calculated using the provided equation, such as \(r = 1 - \cos \theta\).

For accurate plotting, it's helpful to calculate \(r\) at critical angles, such as \(\pi/2\), \(\pi\), \(3\pi/2\), and \(2\pi\). After computing the values, plot the points on polar graph paper or using software. Connect these points to visualize the curve. It is essential in steps like this to pay attention to features such as cusps and symmetry, which guide the comprehensiveness of your graphing approach. Through practice, graphing polar curves becomes intuitive, revealing intricate and aesthetically pleasing shapes.

Symmetry in Polar Curves

Symmetry in polar curves is an intricate concept but offers simple tests to determine how symmetry manifests. In polar equations, symmetry about the x-axis, y-axis, or the origin can simplify the graphing process. A curve is symmetric about the x-axis if replacing \(\theta\) with \(-\theta\) produces the same equation. For symmetry about the y-axis, replace \(r\) with \(-r\).

For a cardioid such as \(r = 1 - \cos \theta\), symmetry about the x-axis is evident, as this equation remains unchanged when \(\theta\) is substituted with \(-\theta\). Similarly, \(r = -1 + \sin \theta\) displays y-axis symmetry, showing no change when mirrored across this axis. Recognizing symmetry helps in reducing errors during plotting and adds to an understanding of the inherent properties that certain polar equations possess, aiding in producing accurate and elegant graphs.