Problem 12 - Precalculus

Question

Graph the polar equations. $$r=1-\sin \theta$$

Step-by-Step Solution

Answer

The graph of \( r = 1 - \sin \theta \) is a cardioid with symmetry about the vertical axis.

Step 1: Understand the Polar Equation

The given polar equation is \( r = 1 - \sin \theta \). In polar coordinates, \( r \) represents the distance from the origin, and \( \theta \) is the angle from the positive x-axis. This type of equation can represent curves like circles, limaçons, and cardioids.

Step 2: Identify the Type of Curve

The form of the equation \( r = 1 - \sin \theta \) suggests it belongs to the family of limaçon curves. When an equation is in the form \( r = a - b\sin \theta \), and if \( a = b \), the curve is typically a cardioid.

Step 3: Analyze the Equation for Key Features

Considering \( r = 1 - \sin \theta \), we find that both \( a \) and \( b \) equal 1. This confirms the curve is a cardioid. The maximum value of \( r \) occurs when \( \sin \theta = -1 \), giving \( r = 1 - (-1) = 2 \). The minimum value of \( r \) occurs when \( \sin \theta = 1 \), giving \( r = 1 - 1 = 0 \).

Step 4: Plot Key Points and Symmetry

Plot points for some key angles: For \( \theta = 0\), \( r = 1 \). For \( \theta = \frac{\pi}{2} \), \( r = 0 \). For \( \theta = \pi \), \( r = 1 \). For \( \theta = \frac{3\pi}{2} \), \( r = 2 \). These points help outline the shape. The equation's symmetry about the vertical axis (since it involves \( -\sin \theta \)) indicates it will look the same on both sides.

Step 5: Sketch the Curve

Begin plotting the graph by marking the origin at \( r = 0 \) for \( \theta = \frac{\pi}{2} \), then move around to extend to \( r = 2 \) at \( \theta = \frac{3\pi}{2} \). Complete the curve by connecting these points smoothly back to \( r = 1 \) at both \( \theta = 0 \) and \( \theta = \pi \). The shape should resemble a heart-like cardioid.

Key Concepts

Polar Equations

Polar equations are a way to describe curves using polar coordinates, which involve a radius and an angle. Unlike the traditional Cartesian system, where points are plotted using x and y coordinates, in polar coordinates each point is indicated by

• the distance from the origin, named \( r \), and
• the angle \( \theta \), measured from the positive x-axis.

This system can represent many shapes effectively, especially those appearing radial or circular in nature. Polar equations help in creating plots that are concentric or spiral, for instance.
For the equation \( r = 1 - \sin \theta \), the value of \( r \) changes as \( \theta \) varies, forming a shape in the polar plane. By understanding how \( \theta \) affects \( r \), you can visualize what shape the equation will generate, such as circles or more complex curves like limaçons and cardioids.

Cardioid

A cardioid is a special type of limaçon, and it shows up as a heart-like shape when plotted. It is a member of the limaçon family of curves, characterized by their distinctive loops and cusps. A cardioid specifically occurs when the parameters of a limaçon equation are equal, such as\( a = b \).
In the equation \( r = 1 - \sin \theta \), because 1 is both \( a \) and \( b \), it forms a cardioid.
The cardioid curve exhibits certain interesting features:

• Its maximum distance from the origin is twice the constant parameter (\( 2 \times 1 = 2 \)).
• The cardioid has one cusp, where the curve meets itself, typically at \( \theta = \pi \over 2 \) for this equation, resulting in \( r = 0 \).
• This shape reflects about the vertical axis due to the \(-\sin \theta \) term, showing symmetry.

Understanding these aspects can help you graph a more accurate depiction of the cardioid in polar coordinates.

Limaçon Curves

Limaçon curves form a broad category of curves in polar equations. These curves are described by equations like \( r = a - b\sin \theta \) or \( r = a - b\cos \theta \). They are fascinating due to their variety:

• If \( a = b \), you get a cardioid, a specific limaçon type.
• When \( a > b \), the limaçon appears as an outer ring or loop, sometimes called a "convex limaçon."
• If \( a < b \), the curve develops an inner loop, becoming an "inner-loop limaçon."

These curves can help illustrate complex mathematical behaviors and are used in various applications, from modeling physics phenomena to art designs. By identifying the values of \( a \) and \( b \), you can anticipate what kind of limaçon the polar equation will form. Knowing whether you'll end up with a cardioid or a more complex loop helps in anticipating the curve's general aesthetic and behavior.