Problem 31 - College Trigonometry

Question

In Exercises \(31-40\), sketch the region in the \(x y\) -plane described by the given set. $$ \\{(r, \theta) \mid 0 \leq r \leq 3,0 \leq \theta \leq 2 \pi\\} $$

Step-by-Step Solution

Answer

The described region is a full circle of radius 3 centered at the origin.

Step 1: Understand Polar Coordinates

In polar coordinates, a point in the plane is determined by a distance, \(r\), from the origin and an angle, \(\theta\), measured from the positive x-axis counterclockwise. For this problem, \(r\) represents the radius and \(\theta\) is the angle.

Step 2: Analyze the Given Set

The given set is \(\{(r, \theta) \mid 0 \leq r \leq 3, 0 \leq \theta \leq 2 \pi\}\). This describes all points where \(r\), the radius, ranges from 0 to 3, and \(\theta\), the angle, goes from 0 to \(2\pi\). This suggests a circular region centered at the origin.

Step 3: Sketch the Circle

To sketch the region, plot a circle with a radius of 3 centered at the origin. Since \(\theta\) encompasses from 0 to \(2\pi\), this forms a full circle. The region includes all points from the center (\(r=0\)) to the boundary of the circle (\(r=3\)).

Step 4: Highlight the Region

Shade the area inside the circle of radius 3, since this includes all radii from 0 to 3. This complete shaded circle is the region described by the polar coordinates set.

Key Concepts

Understanding Radius and Angle in Polar Coordinates

In polar coordinates, every point in the plane is uniquely described by two components: the radius \(r\) and the angle \(\theta\). The radius \(r\) is the distance from the point to the origin. It captures how far out from the center the point is located. This is similar to the concept of a radius in standard circles. The angle \(\theta\) indicates the rotation from the positive x-axis, moving counterclockwise.
A key aspect of working with polar coordinates is visualizing how the radius and angle dynamically change and define locations. For instance:

• A radius \(r = 0\) places the point at the origin, regardless of the angle.
• An angle \(\theta = 0\) aligns the direction with the positive x-axis.
• As \(r\) increases, the point moves further from the origin.

The relationship between the radius and the angle aids in painting a clear picture of any point's placement in the polar system.

Understanding Circular Regions in Polar Coordinates

The concept of a circular region is pivotal when working in polar coordinates. In the given exercise, the set \(\{(r, \theta) \mid 0 \leq r \leq 3, 0 \leq \theta \leq 2\pi\}\) defines a complete circle.
This circular region includes all possible points where:

• The radius \(r\) varies from 0 to 3, sweeping outwards from the origin.
• The angle \(\theta\) moves from 0 to \(2\pi\), covering a full 360 degrees.

The geometry of such a set is easy to visualize as a full circle with radius 3.
Since \(\theta\) reaches from 0 to \(2\pi\), every possible direction from the origin is covered, meaning the entire circular region is filled in.

Reading a Polar Graph Illustrating Regions

A polar graph effectively displays regions defined by polar coordinates. When constructing such a graph, the focus is on the radial distance and angular coverage. Let's take the example from the exercise:
With the radius stretching from 0 to 3 and the full angle from 0 to \(2\pi\), the result is a filled circle. The center of this circular region is at the origin, with every point within the radius depicted.
Here’s how you can approach sketching:

• Draw a circle centered at the origin.
• Ensure the circle's radius is set at 3, encompassing all directions as \(\theta\) covers \(2\pi\).
• Shade the region inside the circle to indicate all points with radius \(0 \leq r \leq 3\).

Such a graph not only shows where points lie but makes it evident how polar coordinates intricately define regions in a plane.