Problem 27 - Precalculus

Question

Graph the polar equations. $$r=4+2 \sin \theta(\text {limacon with no inner loop})$$

Step-by-Step Solution

Answer

The limacon has a dimple and is symmetric about the polar axis.

Step 1: Understanding the Polar Equation

The given polar equation is \(r = 4 + 2 \sin \theta\). This is a limacon equation. Limacons are a family of polar graphs that can have an inner loop, a dimple, or be convex. The form \(r = a + b \sin \theta\) produces different shapes based on the ratio \(b/a\). Here, \(a = 4\) and \(b = 2\).

Step 2: Determine the Type of Limacon

To determine the type of limacon, calculate the ratio \(b/a = 2/4 = 0.5\). If \(0 < b/a < 1\), the limacon has no inner loop and has a dimple. Thus, \(r = 4 + 2 \sin \theta\) will produce a limacon with a dimple.

Step 3: Identify Key Points and Symmetry

Evaluate the equation at key angles to plot points and understand the graph's behavior. For symmetry, note that the equation is symmetric about the vertical line (polar axis symmetry).

Step 4: Calculate Points for Selected Angles

Calculate \(r\) for several values of \(\theta\). For example:- \(\theta = 0\): \(r = 4 + 2 \sin 0 = 4\)- \(\theta = \pi/2\): \(r = 4 + 2 \sin(\pi/2) = 6\)- \(\theta = \pi\): \(r = 4 + 2 \sin \pi = 4\)- \(\theta = 3\pi/2\): \(r = 4 + 2 \sin(3\pi/2) = 2\)

Step 5: Sketch the Graph Based on Calculations

Plot the points calculated for \(\theta = 0, \pi/2, \pi, 3\pi/2\) and observe how \(r\) varies as \(\theta\) changes from \(0\) to \(2\pi\). Draw the limacon shape which should have a dimple towards the direction of the maximum \(r\) value.

Key Concepts

Limacon

In polar coordinates, a limacon is a special type of curve that appears when a point on the plane is expressed using equations like \( r = a + b \sin \theta \) or \( r = a + b \cos \theta \). Limacons can be quite varied in their shape.
Their appearance depends largely on the relationship between the constants \(a\) and \(b\):

• If \(b/a > 1\), you get a limacon with an inner loop.
• If \(b/a = 1\), it resembles a cardioid.
• If \(0 < b/a < 1\), it doesn’t loop and instead takes on a dimpled shape.
• If \(b = 0\), it simplifies to a circle.

In the example equation \(r = 4 + 2 \sin \theta\), the ratio \(b/a = 0.5\), indicating a limacon with a dimple. Understanding these relationships allows one to predict the general shape of the limacon before plotting.

Polar Graph Symmetry

Symmetry plays a crucial role in understanding and graphing polar equations. For polar equations, symmetry makes drawing and analyzing the graph much simpler. Here's how:
For the equation \(r = a + b \sin \theta\):

• Polar Axis Symmetry: Reflects across the polar (horizontal) axis. This can be checked by replacing \(\theta\) with \(-\theta\).
• Line \(\theta = \frac{\pi}{2}\) Symmetry: Reflection symmetry across the vertical line. Substitute \(\theta\) with \(\pi - \theta\).
• Origin Symmetry: The graph reflects through the origin by replacing \(r\) with \(-r\) and \(\theta\) with \(\pi + \theta\).

For the specific limacon \(r = 4 + 2 \sin \theta\), it exhibits symmetry about the vertical line. Recognizing symmetry allows for easier plotting and verification of coordinates.

Trigonometric Functions

Trigonometric functions form the backbone of polar equations. In the polar equation \(r = 4 + 2 \sin \theta\), the trigonometric function \(\sin \theta\) influences the shape and direction of the limacon.
Key aspects of the sine function include:

• Amplitude and Range: The sine function oscillates between -1 and 1. This range influences the overall size of the limacon's dimple or loop.
• Phase and Shift: In the equation, the constants modify how the sine function influences \(r\).

Understanding how the sine function contributes helps anticipate changes in the radius as \(\theta\) varies. This is essential in plotting accurate polar graphs.

Graphing Polar Coordinates

Learning to graph polar coordinates involves converting a polar equation into visual points and curves. The idea is simple: for varying angles \(\theta\), determine \(r\) using the polar equation.
Here are the steps in creating a polar graph:

• Choose Key Angles: These are often \(0\), \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\).
• Calculate \(r\): Substitute \(\theta\) values into the equation to find \(r\). For instance, for \(\theta = 0\), \(r = 4\).
• Plot the Points: Translate each \(r, \theta\) into Cartesian coordinates or directly in polar axes.
• Draw the Curve: Connect the points smoothly, considering symmetry and pattern changes.

Through this process, you’ll achieve an accurate representation of a polar graph, capturing the essence of shapes like the limacon.