Problem 17 - Precalculus

Question

Graph the polar equations. $$r=2+4 \cos \theta$$

Step-by-Step Solution

Answer

The graph of the equation is a limaçon with an inner loop.

Step 1: Convert to Cartesian coordinates

Polar equations are generally in terms of \( r \) and \( \theta \), but when graphing, it is often useful to convert these to Cartesian coordinates \( (x, y) \). The relations are \( x = r \cos \theta \) and \( y = r \sin \theta \). Here, the equation is \( r = 2 + 4 \cos \theta \), so the part to convert is the \( \cos \theta \) in polar form: \( r \cos \theta = x \), leading to the equation \( r - 4x = 2 \).

Step 2: Explore the graph shape

Start by considering the form of the equation. The equation \( r = 2 + 4 \cos \theta \) represents a limaçon. This is a specific type of polar plot that has a distinct loop when \( 2 < 4 \), resulting in a dimpled heart-like shape.

Step 3: Determine key points

Evaluate the equation for some key angles: for \( \theta = 0 \), \( r = 6 \); for \( \theta = \pi/2 \), \( r = 2 \); for \( \theta = \pi \), \( r = -2 \); and for \( \theta = 3\pi/2 \), \( r = 2 \). These values help locate critical points on the graph.

Step 4: Sketch the graph

Using the calculated points and understanding of the limaçon shape, plot these radial distances from the origin. Connect the points with a smooth curve to form the limaçon, noting the loop formed inside as \( r \) becomes negative for \( \pi/2 < \theta < 3\pi/2 \).

Step 5: Verify the loop

Calculate the values at angles where the loop's transition happens, such as \( \theta = \pi/3 \) and \( 2\pi/3 \), to ensure closeness of the loops' size and ensure the graph qualitatively matches expectations.

Key Concepts

Limaçon

The limaçon, a charming mathematical figure, is a type of polar curve. It is famous for its characteristic shape, which can either have a loop or a dimple, giving it a distinctive heart-like or cardiod appearance. A limaçon can be formed from a polar equation of the type \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). Depending on the values of \(a\) and \(b\), the graph of a limaçon can have different shapes:

• A looped limaçon when \( |b| > |a| \).
• A dimpled limaçon when \( |a| > |b| > 0 \).
• A cardioid when \( |a| = |b| \).

In this exercise, we have \( r = 2 + 4 \cos \theta \). Here, \(|4| > |2|\), indicating a looped limaçon. This means that part of the curve loops back on itself, creating an internal loop.

Cartesian coordinates

When working with polar equations, it helps to convert them into Cartesian coordinates for easier graphing. Polar coordinates involve a radial distance, \( r \), and an angle, \( \theta \). By contrast, the Cartesian system uses perpendicular axes, \( x \) and \( y \). These systems are related through the equations \( x = r \cos \theta \) and \( y = r \sin \theta \). For the given polar equation, \( r = 2 + 4 \cos \theta \), there is a need to express it using Cartesian variables. To start, substitute \( r \cos \theta = x \) into the equation, transforming it into \( r - 4x = 2 \). This transformation provides a new perspective on the graph, letting us see key geometric features that might be less visible in the polar form.

Radial distances

Radial distance plays a crucial role in understanding and sketching polar equations like that of a limaçon. Radial distance, \( r \), represents how far a point is from the origin (the pole) in the polar coordinate system. In our example \( r = 2 + 4 \cos \theta \), calculate \( r \) at specific values of \( \theta \) to find pivotal points on the graph. For instance:

• When \( \theta = 0 \), \( r = 6 \).
• When \( \theta = \pi/2 \), \( r = 2 \).
• When \( \theta = \pi \), \( r = -2 \).
• When \( \theta = 3\pi/2 \), \( r = 2 \).

The negative \( r \) value at \( \theta = \pi \) is essential: it indicates the position along the opposite direction, critical for the limaçon's interior loop. Ensuring accurate radial distances at these angles helps build a precise graphical representation.

Graphing techniques

To graph polar equations like \( r = 2 + 4 \cos \theta \), a structured approach using particular techniques simplifies the process. Begin by identifying key points, utilize symmetry, and understand the behavior of \( r \) as \( \theta \) changes. This step-by-step strategy includes:

• **Choosing key angles:** Evaluate the equation for angles like \( 0 \), \( \pi/2 \), \( \pi \), and \( 3\pi/2 \) to pinpoint crucial graph junctures.
• **Using symmetry:** Many polar graphs exhibit symmetry, which helps in accurately sketching the entire plot.
• **Plotting radial distances:** Mark the calculated radial distances from the origin in their respective directions.
• **Connecting points smoothly:** Draw a continuous and smooth curve through these points, considering the equation's structural traits.

Understanding the nature of the plot, especially features like loops in a limaçon, requires calculating and verifying points around potential transitions or changes in curvature. This ensures that the final graph adheres closely to theoretical expectations.