Problem 41 - Algebra and Trigonometry

Question

Identify and graph each polar equation. $$ r=3-3 \sin \theta $$

Step-by-Step Solution

Answer

The equation \(r = 3 - 3 \sin \theta\) represents a cardioid. Plot the points calculated at key angles to graph it.

Step 1: Identify the Polar Equation Form

The given polar equation is in the form of a limaçon: \[ r = a \,±\, b\, \text{sin} \theta \] Here, we have: \[ r = 3 - 3 \, \text{sin} \theta \] This identifies it as a limaçon with values \(a = 3\) and \(b = 3\).

Step 2: Determine the Type of Limaçon

Since \(a = b\), the equation represents a cardioid. A cardioid is a specific type of limaçon.

Step 3: Find Key Points

Calculate key points by substituting \(\theta\) values (0, \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\)) into the equation: \(\theta = 0: r = 3 - 3 \cdot \sin(0) = 3 - 0 = 3\) \(\theta = \frac{\pi}{2}: r = 3 - 3 \cdot \sin(\frac{\pi}{2}) = 3 - 3 = 0\) \(\theta = \pi: r = 3 - 3 \cdot \sin(\pi) = 3 - 0 = 3\) \(\theta = \frac{3\pi}{2}: r = 3 - 3 \cdot \sin(\frac{3\pi}{2}) = 3 - (-3) = 6\)

Step 4: Plot the Graph

Use the key points to plot the graph in polar coordinates. Connect these points smoothly to form the cardioid shape. Remember, at \(\theta = \frac{\pi}{2}\), the point is at the origin. At \(\theta = \frac{3\pi}{2}\), the point is the farthest from the origin with a distance of 6 units.

Key Concepts

Limaçon

A limaçon is a unique type of polar curve described by the equation \( r = a \,± \, b \, \text{sin} \, \theta \) or \( r = a \,± \, b \, \text{cos} \, \theta \). These types of curves can take on different shapes depending on the relationship between the constants \( a \) and \( b \).
The most common types of limaçon curves include:

• A dimpled limaçon, which appears when \( a > b \)
• A looped limaçon, occurs when \( a < b \)
• A cardioid, forms specifically when \( a = b \)

This variety makes studying limaçon curves especially interesting, as they can illustrate a range of geometric properties within polar coordinates.

Cardioid

A cardioid, a special type of limaçon, is characterized by its heart-like shape. The polar equation for a cardioid is \( r = a \, ± \, a \, \text{sin} \, \theta \) or \( r = a \, ± \, a \, \text{cos} \, \theta \). When plotting a cardioid:

• Ensure \( a = b \), making the leading coefficients equal.
• Notice the curve passes through the origin, giving it a unique symmetry.
• Cardioids have a central cusp, providing its distinct heart shape.

In the example \( r = 3 - 3 \, \text{sin} \, \theta \), the coefficients are identical, confirming it is a cardioid.

Polar Coordinates

Polar coordinates offer an alternative way to describe and graph points in the plane using angles and distances. Instead of the Cartesian \( (x, y) \) system, polar coordinates identify a point by its distance from the origin and its angle from the positive x-axis. The notation \( (r, \, \theta) \) is used, where

• \( r \) represents the radial distance from the origin
• \( \theta \) represents the angular coordinate (usually measured in radians)

To convert between polar and Cartesian coordinates:

• Use \( x = r \, \text{cos} \, \theta \)
• Use \( y = r \, \text{sin} \, \theta \)

Understanding polar coordinates is key for graphing equations like the limaçon and cardioid.

Graphing Polar Equations

Graphing polar equations involves plotting points based on their polar coordinates \( (r, \, \theta) \) and connecting these points to visualize the curve. Follow these steps for effective graphing:

• Identify the polar equation form (e.g., limaçon, cardioid).
• Substitute important angle values (\( \theta = 0, \, \frac{\text{π}}{2}, \, \text{π}, \, \frac{3\text{π}}{2} \)) to find the corresponding \( r \) values.
• Plot the points on polar coordinate paper.
• Connect the points smoothly to complete the curve.

For the given example \( r = 3 - 3 \, \text{sin} \, \theta \), the values of \( r \) at these angles were
0, 3, and 6, showing a complete and symmetrical cardioid graph.