Problem 33 - Algebra and Trigonometry

Question

15–36 Sketch the graph of the polar equation. $$r=2+\sin \theta \quad \text {(limacon)}$$

Step-by-Step Solution

Answer

Graph a cardioid with minimum \( r = 1 \) and maximum \( r = 3 \) based on \( r = 2 + \sin \theta \).

Step 1: Understand the Polar Equation

The given polar equation is \( r = 2 + \sin \theta \). This is a type of limacon because it is of the form \( r = a + b \cdot \sin \theta \), where both \( a \) and \( b \) are equal to 2.

Step 2: Identify Key Features

Since \( a = b = 2 \), the limacon is a cardioid. Cardioids occur when the absolute values of \( a \) and \( b \) are equal. The graph will have a 'heart' shape.

Step 3: Determine Range of r Values

The minimum value of \( r \) is \( a - b = 2 - 1 = 1 \) and the maximum value is \( a + b = 2 + 1 = 3 \). Thus, \( r \) ranges from 1 to 3.

Step 4: Calculate Key Points

Calculate \( r \) for critical angles: \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \). At \( \theta = 0, r = 2 + \sin(0) = 2 \). At \( \theta = \frac{\pi}{2}, r = 2 + \sin(\pi/2) = 3 \). At \( \theta = \pi, r = 2 + \sin(\pi) = 2 \). At \( \theta = \frac{3\pi}{2}, r = 2 + \sin(3\pi/2) = 1 \).

Step 5: Sketch the Graph

Plot the points calculated in Step 4 in polar coordinates. Mark the minimum value \( r = 1 \) on the negative y-axis and the maximum value \( r = 3 \) on the positive y-axis. Connect the points with a smooth curve to complete the cardioid shape.

Key Concepts

Limacon

A limacon is a type of curve that can be described using polar coordinates. It is often represented by the equation \(r = a + b \cdot \sin \theta\) or \(r = a + b \cdot \cos \theta\). The graph of a limacon is unique and varies depending on the relationship between the constants \(a\) and \(b\). Here are some important points about limacons:

• A limacon can appear as a simple loop, a dimpled loop, or as a cardioid.


• The shape depends on whether \(a < b, a = b,\) or \(a > b\).


• In our example, \(a = b = 2\), forming a cardioid.

Understanding limacons helps you to identify the type of curve you are dealing with and anticipate the graph's characteristics.

Polar Coordinates

Polar coordinates offer a different way to describe the location of a point, utilizing the distance from a fixed point and an angle from a fixed direction. Unlike Cartesian coordinates (\(x, y\)), polar coordinates are expressed as \( (r, \theta) \). Here's how it works:

• \(r\) is the distance from the origin to the point.


• \(\theta\) is the angle measured from the positive x-axis in the counter-clockwise direction.

In polar graphs, curves are plotted using values of \(r\) and angles \(\theta\). This is especially useful for plotting spirals and circular shapes, such as limacons and cardioids.
The flexibility of polar coordinates makes it easier to understand and sketch complex curves.

Cardioid

A cardioid is a special type of limacon where \(a = b\). Named for its resemblance to a heart shape, the cardioid's equation in polar coordinates is typically \(r = a + a \cdot \sin \theta\) or \(r = a + a \cdot \cos \theta\). Here are some interesting facts about cardioids:

• They have a single cusp and are entirely symmetric.


• Both halves of the shape can be folded against each other.

For the equation \(r = 2 + \sin \theta\), a cardioid appears because both constants are equal, which means the graph will be symmetric about the polar axis. By plotting key points at specific angles, such as 0, \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\), you can map the full cardioid shape on a polar graph.

Trigonometric Functions

Trigonometric functions like \(\sin\), \(\cos\), and \(\tan\) are foundational in graphing polar equations. These functions help define how the radius \(r\) changes with the angle \(\theta\). Here's how they relate to our problem:

• In the equation \(r = 2 + \sin \theta\), the \(\sin \theta\) term determines the variation of \(r\) as \(\theta\) changes.


• \(\sin \theta\) reaches its maximum at \(\theta = \frac{\pi}{2}\) and minimum at \(\theta = \frac{3\pi}{2}\).

Understanding these variations is key to identifying critical points on the graph.
For instance, when \(\theta = \frac{\pi}{2}\), \(r\) reaches its maximum of 3 in our example. These trigonometric relationships help in accurately sketching the polar graph and predicting the shape it will take.