Problem 29 - Thomas Calculus

Question

Which of the following has the same graph as \(r=1-\cos \theta ?\) a. \(r=-1-\cos \theta \quad\) b. \(r=1+\cos \theta\) Confirm your answer with algebra.

Step-by-Step Solution

Answer

Option b, \(r = 1 + \cos \theta\), has the same graph type as \(r = 1 - \cos \theta\).

Step 1: Understand the Graph of the Polar Equation

The original equation given is \(r = 1 - \cos \theta\). This is the equation of a cardioid, which is a special type of limaçon that has a cusp at \(\theta = 0\). The curve lies in the right half-plane.

Step 2: Analyze Option a

Option a presents the equation \(r = -1 - \cos \theta\). We can rewrite this to resemble a form we understand, thus: \(-r = 1 + \cos \theta\), which implies \(r = -(1 + \cos \theta)\). The negative sign indicates that for any \(\theta\), \(r\) takes the opposite sign, placing the curve in the opposite quadrant compared to original \(r = 1 - \cos \theta\). It does not match the original graph.

Step 3: Analyze Option b

Option b presents the equation \(r = 1 + \cos \theta\). This equation is another form of a cardioid. By comparing \(r = 1 + \cos \theta\) with \(r = 1 - \cos \theta\), the critical difference is the sign in front of \(\cos \theta\). However, \(r = 1 + \cos \theta\) results in the graph being a shifted cardioid, but both are cardioids and have similar characteristics.

Step 4: Confirm with Algebra

A precise algebraic confirmation might consider expressing \(r = 1 - \cos \theta\) in Cartesian coordinates or examining specific points, but due to symmetry, cardioids \(r = 1 \pm \cos \theta\) have their cusps along the horizontal axis and otherwise retain similar shapes, confirming similarity in graphs.

Key Concepts

cardioid

A cardioid is a fascinating shape that you might encounter when exploring polar coordinates. It is named for its heart-like appearance. In polar equations, a cardioid can be identified by equations of the form:

• \( r = a(1 \pm \cos \theta) \)
• \( r = a(1 \pm \sin \theta) \)

Here, \( a \) is usually a positive constant.
The presence of either cosine or sine suggests orientation along particular axes:

• \( r = 1 - \cos \theta \) is a cardioid that typically traces the right half of the plane, with its cusp touching the origin.
• \( r = 1 + \cos \theta \) similarly forms a cardioid, but its defining shift gives it a distinctive shape on the right half-plane.

Remember, the peculiarity of a cardioid is that it has one cusp where it touches the coordinate pole.

limaçon

The limaçon is a versatile curve which can manifest different appearances including a dimple, a loop, or even a cardioid. It takes its name from the French word for "snail," reflecting its often spiral-like shape.
The general polar equation for a limaçon is:

• \( r = a + b \cos \theta \)
• \( r = a + b \sin \theta \)

Depending on the relationship between \( a \) and \( b \), the limaçon changes its form:

• When \( |a| = |b| \), the limaçon becomes a cardioid, showing off its heart-like contour.
• If \( |a| > |b| \), the curve has a dimple but no inner loop.
• Conversely, if \( |a| < |b| \), a loop appears, making the curve more complex.

In the case of the problem, both \( r = 1 - \cos \theta \) and \( r = 1 + \cos \theta \) are cardioids. This exemplifies how closely related cardioids and limaçons are.

cusp

A cusp is a unique point on a curve where the direction changes abruptly, like a sharp corner or a peak. In polar equations, a cardioid has a cusp located at a specific angle, which is usually evident when examining its graph.
For the cardioid described by \( r = 1 - \cos \theta \), the cusp appears at:

• \( \theta = 0 \), where the curve makes contact with the origin.

This is a result of the shape folding over itself at this point, creating a visibly pointed feature.
In general, a cusp in a polar graph can signify where the radius \( r \) either becomes zero or reaches a minimal extent. Knowing the presence of a cusp can help you visualize the overall symmetry and behavior of the curve, especially for cardioids. Such cusps are also a key distinguishing factor between the different forms a limaçon can take.