Question

Identify and graph each polar equation. $$ r=4+2 \sin \theta $$

Short answer

The curve is a Limaçon with parameters \(a = 4\) and \(b = 2\). It has maximum and minimum radii of 6 and 2, respectively.

Step-by-step solution

Convert Polar Equation for Understanding

The polar equation given is \(r = 4 + 2 \sin \theta\). Understand that this represents a polar curve where \(r\) (the radius) varies depending on the angle \(\theta\). This equation is better understood as a sum of a constant radius and a sinusoidal component.

Identify the Type of Curve

Recognize that this equation represents a Limaçon curve. A Limaçon curve is of the form \(r = a + b \sin \theta\), where \(a\) and \(b\) are constants. Therefore, the given curve is a Limaçon with parameters \(a = 4\) and \(b = 2\).

Analyze the Parameters

For \(r = 4 + 2 \sin \theta\), identify that the maximum radius occurs when \(\sin \theta\) is 1, giving \(r = 4 + 2 = 6\). The minimum radius occurs when \(\sin \theta\) is -1, giving \(r = 4 - 2 = 2\). The value of \(a\) (4) determines the overall size and shift of the Limaçon.

Create a Table of Values

To plot the graph, create a table of values for \(\theta\) from 0 to 2\(\pi\) in increments (e.g., 0, \(\pi/6\), \(\pi/4\), ...), and calculate the corresponding \(r\) values. For instance: \(\theta = 0\), \(r = 4\); \(\theta = \pi/2\), \(r = 6\); \(\theta = \pi\), \(r = 4\); \(\theta = 3\pi/2\), \(r = 2\); etc.

Plot the Points on Polar Grid

Plot these \((\theta, r)\) points on a polar grid. Ensure that every angle \(\theta\) corresponds to its radius \(r\). Use the calculated points to understand the shape of the curve. Note how the radius changes with \(\theta\).

Connect the Points to Form the Curve

Once multiple points are plotted, connect these points smoothly to reveal the shape of the Limaçon curve. Notice the characteristic 'flattened' bottom when \(\theta\) is around 3\(\pi/2\) (reflecting the minimum \(r\)).

Key concepts

Limaçon Curve

The Limaçon curve is a special type of polar curve, characterized by its unique heart-like shape.
These curves are defined by equations of the form \(r = a + b \sin \theta\). In the given equation, \(r = 4 + 2 \sin \theta\), we see that \(a = 4\) and \(b = 2\). This indicates that the curve is a Limaçon.
Depending on the values of \(a\) and \(b\), the Limaçon can look different:

• If \(|a| > |b|\), the Limaçon has an outer loop.
• If \(|a| = |b|\), it forms a cardioid.
• If \(|a| < |b|\), there is an inner loop.

In our case, since \(|a| > |b|\) with \(a = 4\) and \(b = 2\), the Limaçon doesn't have a loop but has a dimple at the bottom. This is reflected in the equation \(r = 4 + 2 \sin \theta\) as \(\r\) gets smaller around \( \theta = 3 \pi / 2 \). Understanding these parameters is key to graphing and analyzing such curves.

Polar Coordinates

Polar coordinates represent a point in the plane using a radius and an angle rather than x and y coordinates. The coordinate (r, \(\theta\)) describes:

• \(r\): The distance from the origin.
• \(\theta\): The angle from the positive x-axis.

In our given problem, \(\r = 4 + 2 \sin \theta\), the value of \(r\) changes as \(\theta\), the angle, changes. These values help plot the points on a polar grid.
Polar coordinates can visually demonstrate sinusoidal variations, such as seen in the equation \(r = 4 + 2 \sin \theta\). When \(\theta\) changes, the value of \(r\) traces out a curve, revealing the overall shape of the graph. Understanding how (r, \(\theta\)) pair works is crucial.

Graphing Techniques

Graphing polar equations like \(r = 4 + 2 \sin \theta\) involves several key techniques:

• Create a Table of Values: Start by calculating \(r\) values for various \(\theta\) angles from \(0 \pi\) to \(2 \pi\).
• Plot Points: Using polar coordinates, plot each \( (\theta, r)\) on a polar grid.
• Connect Points: Smoothly connect these points to reveal the Limaçon curve. The radius increases and decreases in a sinusoidal manner, showing dimple at the bottom when \( \theta = 3 \pi /2 \).

By using these graphing techniques, you can effectively visualize how the polar equation \(r = 4 + 2 \sin \theta\) translates into the 2D plane and forms a meaningful curve. Understanding these steps ensures accurate graphing and helps in drawing insights about the behavior of polar equations.