Problem 46 - College Trigonometry

Question

In Exercises \(41-50\), use set-builder notation to describe the polar region. Assume that the region contains its bounding curves. The region inside the cardioid \(r=2-2 \sin (\theta)\) which lies in Quadrants I and IV.

Step-by-Step Solution

Answer

The region is \( \{(r, \theta) \mid 0 \leq r \leq 2-2\sin(\theta), \theta \in [0, \frac{\pi}{2}] \cup [\frac{3\pi}{2}, 2\pi]\} \).

Step 1: Understanding the Cardioid Equation

The cardioid given by the equation \(r=2-2\sin(\theta)\) describes a curve in the polar coordinate system. Analysis of this equation shows that \(\theta\) ranges from \(0\) to \(2\pi\), tracing out a heart-shaped curve (a cardioid) symmetric about the horizontal axis (polar axis).

Step 2: Identifying the Quadrants of Interest

Quadrants I and IV correspond to angles from \(0\) to \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\) to \(2\pi\), respectively, in the polar coordinate system.

Step 3: Determining the Bounding Curves

Since the entire cardioid must be covered, our region is defined by the inequality \(0 \leq r \leq 2-2\sin(\theta)\) for the specified \(\theta\).

Step 4: Expressing the Region in Set-Builder Notation

We define the region using set-builder notation. The region \(R\) inside the cardioid is given by: \[ R = \{(r, \theta) \mid 0 \leq r \leq 2-2\sin(\theta), \theta \in [0, \frac{\pi}{2}] \cup [\frac{3\pi}{2}, 2\pi]\} \].

Key Concepts

Cardioid Equation

A cardioid is a special type of curve that you'll often encounter when dealing with polar coordinates. The equation of a cardioid typically appears in the form of

• \( r = a \pm a \sin(\theta) \) or
• \( r = a \pm a \cos(\theta) \).

In this exercise, we encounter a cardioid described by the equation

• \( r = 2 - 2\sin(\theta) \),

where the constants determine the size and orientation of the cardioid.
The term

• \( 2 \) indicates that the cardioid is scaled by a factor of 2.
• The \( -2\sin(\theta) \) causes it to be oriented symmetrically about the polar axis, giving it that distinct heart shape.

A cardioid has unique properties, including being naturally curvy and symmetrical about either the horizontal or vertical axis, depending on whether the sine or cosine function is involved.
These shapes are best visualized when graphed, but it's essential to understand that every point on a cardioid corresponds to a specific radial distance \( r \) for a particular angle \( \theta \), making polar coordinates an effective way to describe such curves.

Set-Builder Notation

Set-builder notation is a concise and efficient way to define a set, and it's especially useful when describing regions within polar coordinates.
In the exercise, the region inside the cardioid is expressed as

• \[ R = \{(r, \theta) \mid 0 \leq r \leq 2-2\sin(\theta), \theta \in [0, \frac{\pi}{2}] \cup [\frac{3\pi}{2}, 2\pi]\} \].

This notation captures all the essential information:

• First, it specifies the variables \((r, \theta)\) involved.
• Second, it provides the condition that defines the set—in this case, the inequality \(0 \leq r \leq 2-2\sin(\theta)\).
• Lastly, it describes the intervals for \(\theta\), noting the particular quadrants the region includes (in this case, Quadrants I and IV).

The use of \(\cup\) denotes a union of sets, which allows us to describe regions that are continuous over specific intervals.
Set-builder notation is powerful as it offers a way to succinctly convey complex regions or areas by using simple mathematical language.

Polar Regions

Polar regions are areas defined by a range of radii \( r \) and angles \( \theta \) in polar coordinates.
They can represent various shapes, such as circles, spirals, or more complex curves like cardioids.
In polar coordinates, polar regions are typically defined by a pair of inequalities:

• One that gives a range for \( r \),
• Another that specifies the permissible angles \( \theta \).

For example, in our exercise, the region we explore is defined by

• \( 0 \leq r \leq 2 - 2\sin(\theta) \) and
• \( \theta \in [0, \frac{\pi}{2}] \cup [\frac{3\pi}{2}, 2\pi] \),

indicating that it encompasses certain arcs of the cardioid's perimeter.
When visualizing polar regions, it's often useful to plot them on the polar grid, where each point is determined by its radial distance from the origin and the direction relative to the positive horizontal axis.
Understanding how to describe and graph polar regions can greatly aid in visualizing and solving polar coordinate problems, making it easier to comprehend areas bounded by curves and lines.