Problem 41 - 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 \(r=5\).

Step-by-Step Solution

Answer

\( \{ (r, \theta) \mid 0 \leq r \leq 5, \theta \in [0, 2\pi) \} \)

Step 1: Identify the polar region

The exercise asks for the region inside the circle defined by the equation \( r = 5 \). In polar coordinates, this includes all points where the distance from the origin to the point (radius \( r \)) is less than or equal to 5.

Step 2: Express the region using inequalities

Since we need the region inside the circle, the radius \( r \) must be less than or equal to 5. Mathematically, this is represented by the inequality \( 0 \leq r \leq 5 \).

Step 3: Express in set-builder notation

Set-builder notation is a way to describe a set by stating the properties that its members must satisfy. For our polar region, the set includes all points \((r, \theta)\) such that the radius \( r \) satisfies \( 0 \leq r \leq 5 \) and \( \theta \) can be any angle (since the circle is complete around the origin). Thus, the set-builder notation is: \[ \{ (r, \theta) \mid 0 \leq r \leq 5, \theta \in [0, 2\pi) \} \]

Key Concepts

Set-builder Notation

Set-builder notation is a concise and mathematical way to describe a set by specifying the properties that members of the set must satisfy. It is particularly useful in specifying complex or infinite sets where listing every element is impractical. When using set-builder notation, the general format is:

• \[ \{ x \mid \, \text{condition involving}\, x \} \]

The symbol \(|\mid|\) can be read as "such that." This means that every element \(x\) in the set satisfies the given condition. For polar coordinates, this extends to pairs \((r, \theta)\).

In the context of polar regions, we can use set-builder notation to define all pairs \((r, \theta)\) that lie within a certain geometric shape, such as a circle or sector. Here, \(r\) represents the distance from the origin, and \(\theta\) represents the angular coordinate. By precisely defining these parameters, we can capture the region mathematically without having to rely on graphical representation alone.

Inequalities in Polar Coordinates

In polar coordinates, inequalities help us define specific regions by setting boundaries on the radial distance \(r\) and angle \(\theta\). Unlike Cartesian coordinates, polar coordinates require consideration of both the angle and the radius.

For example, when focusing on the region inside a circle defined by \(r = a\), we use the inequality \(0 \leq r \leq a\). This conveys that all points must be at a radius less than or equal to \(a\). For angles, since polar coordinates often complete a full rotation around the origin, \(\theta\) usually ranges from \([0, 2\pi)\), unless restricted by other conditions.

• Inner circle or disk: \(0 \leq r \leq 5\) captures a disk with radius 5.
• Restricted angles in sectors: involves inequalities on \(\theta\).

Using inequalities lets us define not only simple circular regions but also more complex shapes, such as annular sections or radial wedges, by further tweaking the limitations on \(r\) and \(\theta\).

Polar Regions

Polar regions refer to specific areas in the plane defined using polar coordinates, \((r, \theta)\). By dictating the range of \(r\) and \(\theta\), we can encapsulate different shapes and areas, like circles, annuli, and sectors.

The polar coordinate system is based around an origin point and an angle emanating from the positive x-axis. This makes it ideal for problems involving circular symmetry, as each point in the region is defined by its distance from the origin and the angle from a fixed direction.

A common example is the complete circle centered at the origin, represented as:

• 
  The radius \(r\) fills in the circle from the origin up to its boundary.
• Any angle \(\theta\) from 0 to \(2\pi)\) completes the surrounding rotation.

This description covers the entire area within the bounds, making polar regions particularly useful in contexts such as physics, engineering, and computer graphics, where natural circular paths and rotations frequently occur.