Problem 39 - A Graphical Approach to Precalculus with Limits

Question

Graph each polar equation for \(\theta\) in \(\left[0^{\circ}, 360^{\circ}\right)\). In Exercises \(39-48\), identify the rype of polar graph. $$r=2+2 \cos \theta$$

Step-by-Step Solution

Answer

The graph is a limacon with an inner loop.

Step 1: Convert polar equation to rectangular

Let's start by converting the given polar equation \(r = 2 + 2 \cos \theta\) to its rectangular form. We know that \(x = r\cos\theta\) and \(y = r\sin\theta\). Thus, \(r^2 = x^2 + y^2\) and \(r = \sqrt{x^2 + y^2}\). Inserting the equation \(r = 2 + 2\cos\theta\), we can express \( \cos\theta \) as \( \frac{x}{r} = \frac{x}{\sqrt{x^2 + y^2}} \). This helps us understand how the polar coordinates relate to rectangular ones.

Step 2: Identify and Simplify

The graph of \( r = 2 + 2 \cos \theta \) is known as a "limacon with an inner loop". A limacon's general form is \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). Depending on the values of \(a\) and \(b\), it may or may not have an inner loop. Since \(a = b = 2\), it falls into the category of a limacon with an inner loop.

Step 3: Determine points for graph

To graph the limacon, set specific angles \(\theta\) to determine \(r\). For example: - \( \theta = 0^{\circ} \Rightarrow r = 2 + 2 \times 1 = 4\) - \( \theta = 90^{\circ} \Rightarrow r = 2 + 2 \times 0 = 2\) - \( \theta = 180^{\circ} \Rightarrow r = 2 - 2 = 0\) - \( \theta = 270^{\circ} \Rightarrow r = 2 + 2 \times 0 = 2\)Repeat similar calculations to determine additional points at other angles.

Step 4: Plot and sketch

Using the determined points, plot them on a polar coordinate graph and connect them to sketch the limacon. Note that the graph will start showing a loop inside the main figure, which confirms it as a limacon with an inner loop. The graph symmetry will help complete the figure.

Key Concepts

Limacon

A limacon is a fascinating type of polar graph that is defined by equations of the form \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). These graphs are famous for their distinctive heart-shaped curves, which can sometimes include an additional inner loop. The shape of a limacon depends on the relationship between the constants \( a \) and \( b \).

• When \( a > b \), the limacon looks somewhat like an egg without an inner loop.
• If \( a = b \), the graph will show a cardiod—a heart shape without an inner loop.
• When \( a < b \), the limacon will have an inner loop, as illustrated by the equation \( r = 2 + 2 \cos \theta \).

To graph a limacon, such as the one described by the equation \( r = 2 + 2 \cos \theta \), it's essential to identify key points by substituting specific angles \( \theta \) into the equation. The result will guide you towards plotting the shape accurately, displaying its distinctive inner loop.

Polar Coordinates

Polar coordinates offer a unique system for graphing points on a plane. Unlike the rectangular coordinate system which uses \( x \) and \( y \) coordinates, polar coordinates use a radius \( r \) and an angle \( \theta \). In this system, each point on a plane is determined by how far it is from the origin and the angle it forms with the positive x-axis.

• The distance from the origin is represented by \( r \), providing a measure of how far the point is from the pole (or center).
• The angle \( \theta \) is typically measured in radians or degrees and indicates the direction from the origin.

One advantage of this system is its ease in representing curves like spirals and circles, which can be cumbersome in rectangular coordinates. For example, the polar equation \( r = 2 + 2 \cos \theta \) helps to express the complex limacon shape in a simplified manner, capturing both its symmetry and unique curve characteristics by varying \( \theta \).

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the most common way to describe points in a plane using the \( x \) and \( y \) axes. Every point in this system is identified by its horizontal (\( x \)) and vertical (\( y \)) distances from the origin, defined by the intersection of the two axes.
To convert polar coordinates \((r, \theta)\) to rectangular coordinates, the equations \( x = r \cos \theta \) and \( y = r \sin \theta \) are used. Additionally, \( r = \sqrt{x^2 + y^2} \) and \( \cos \theta = \frac{x}{r} \) help in translating equations back from polar form to rectangular form as needed.

• Ease of plotting: Rectangular coordinates make it straightforward to plot linear and parabolic curves.
• Comparison with polar coordinates: While rectangular coordinates excel in certain mathematical contexts, polar coordinates maintain advantages in graphing circular and spiral shapes.

Understanding how to transition between polar and rectangular coordinates is crucial, especially when interpreting complex shapes like the limacon in both coordinate systems. This conversion enriches comprehension, offering different perspectives and analysis methods for graphing challenges.