Problem 48 - 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 circle \(r=5\) but outside the circle \(r=3\).

Step-by-Step Solution

Answer

The region is \(\{ (r, \theta) \mid 3 \leq r \leq 5, \; 0 \leq \theta < 2\pi \}\).

Step 1: Identify the Region

The problem asks us to describe a polar region using set-builder notation. The region is inside the circle with radius 5, described by the inequality \(r \leq 5\), and outside the circle with radius 3, described by \(r \geq 3\).

Step 2: Define Set in Polar Coordinates

In polar coordinates, we describe the region where the radius \(r\) satisfies \(3 \leq r \leq 5\). Meanwhile, the angular coordinate \(\theta\) can range from \(0\) to \(2\pi\), since no restrictions are given.

Step 3: Construct the Set-Builder Notation

Using the inequalities developed, construct the set in set-builder notation: \(\{ (r, \theta) \mid 3 \leq r \leq 5, \; 0 \leq \theta < 2\pi \}\). This notation describes the region that contains points where the radius is between 3 and 5, and the angle \(\theta\) covers the entire circular range.

Key Concepts

Polar Coordinates

Polar coordinates are an essential part of mathematics and are particularly useful for describing the location and region of points within a plane. Unlike Cartesian coordinates, which use a grid of horizontal and vertical lines, polar coordinates employ a different approach. They specify a point's location based on two values:

• The radial coordinate, denoted as \( r \) – which is the distance from the pole, or origin, to the point.
• The angular coordinate, denoted as \( \theta \) – which is the angle formed with the positive x-axis.

The radial coordinate \( r \) tells us how far away the point is from the origin, while \( \theta \) gives us the direction or angle.
In this exercise, we've seen that points within a certain region can be described using these coordinates. For instance, the region inside one circle and outside another can be clearly outlined using inequalities in relation to \( r \). This makes polar coordinates particularly powerful when dealing with circular boundaries.

Inequalities

Inequalities are a fundamental part of mathematics that allow us to express the range of values that a variable can assume. In the context of polar coordinates, inequalities help define regions based on distance and direction.
For the given problem, we encounter two inequalities:

• \( r \leq 5 \), which means the radial distance \( r \) should be less than or equal to 5, signifying the space inside a circle with radius 5.
• \( r \geq 3 \), which indicates that \( r \) should be greater than or equal to 3, representing the area outside a circle with radius 3.

By combining these, we constrict \( r \) to be between 3 and 5, \( 3 \leq r \leq 5 \), effectively describing the annular region or ring between the two circles.
Inequalities thus allow us to precisely dictate where points are located relative to another point or boundary, essential for defining regions in mathematics.

Polar Region Description

A polar region description often involves specifying an area or a region using inequalities and polar coordinates. It is a vital concept in problems where describing complex shapes or areas using traditional Cartesian equations may appear complicated.
In this exercise, we are tasked with describing a specific polar region using set-builder notation. The region is essentially an annular ring located between two concentric circles with different radii. The region can be expressed using inequalities that combine both radial and angular components:

• The radial constraint \( 3 \leq r \leq 5 \) delineates the space between and including the two circles.
• The angular component \( 0 \leq \theta < 2\pi \) indicates that there are no restrictions on the angle of the points in this region, covering a full circle.

Combining these elements into set-builder notation formalizes the solution: \( \{ (r, \theta) \mid 3 \leq r \leq 5,\; 0 \leq \theta < 2\pi \} \). This concise expression completely describes the polar region of interest, allowing one to understand the boundaries and range of points contained within this area. This method not only provides clarity in representing regions but also highlights the utility of combining polar coordinates with set-builder notation in mathematical exercises.