Problem 42 - 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 \(r=5\) which lies in Quadrant III.

Step-by-Step Solution

Answer

Set in set-builder notation: \( \{ (r, \theta) \mid 0 \leq r \leq 5, \pi \leq \theta \leq \frac{3\pi}{2} \} \).

Step 1: Identify the Bounds in Polar Coordinates

The problem specifies a circle with radius 5, meaning that within polar coordinates the set we are looking at is all points that satisfy: \[0 \leq r \leq 5\]

Step 2: Recognize Quadrant III in Polar Coordinates

In polar coordinates, Quadrant III corresponds to angles that range from \(\pi\) to \(\frac{3\pi}{2}\). This means the partial region of the circle we are interested in has angles: \[\pi \leq \theta \leq \frac{3\pi}{2}\]

Step 3: Construct the Set Using Set-Builder Notation

Combine the bounds for radius and angle to describe the region in set-builder notation.The set of all points within this region is:\[\{ (r, \theta) \mid 0 \leq r \leq 5, \pi \leq \theta \leq \frac{3\pi}{2} \}\]

Step 4: Verify the Region Description

This set describes all points inside a circle of radius 5 centered at the origin. Additionally, it confirms that only points in Quadrant III are considered through the specified range of \(\theta\). Thus, the description matches the specified region.

Key Concepts

Polar Coordinates

Polar coordinates offer a way to locate points in a plane using a radius and an angle rather than the traditional x and y layout of Cartesian coordinates. In polar coordinates, each point on the plane is represented as

• \((r, \theta)\), where:
• \(r\) is the distance from the origin (a non-negative number), and
• \(\theta\) is the angle formed with the positive x-axis, typically measured in radians.

This system is especially helpful for problems involving circular or rotational symmetry. Unlike Cartesian coordinates, where each point has a unique representation, in polar coordinates, multiple \((r, \theta)\)pairs can represent the same point by adding 2\(\pi\) to \(\theta\) or using negative values for \(r\). This flexibility makes polar coordinates ideal for describing circles and other curves with rotational characteristics.

Quadrant III

In the context of polar coordinates, the plane is divided into 4 quadrants, much like in Cartesian coordinates, but they are described with angles. Quadrant III is where both the x and y coordinates are negative in Cartesian terms. In polar coordinates, this quadrant corresponds to:

• Angles from \(\pi\) to \(\frac{3\pi}{2}\), which places the line in the lower half of the plane extending leftwards.

Points in Quadrant III usually appear as negative x and y values in Cartesian terms but positive radius values in polar terms, highlighted by the specified angle range. This distinction is crucial when describing regions using polar coordinates as it impacts the angle values selected to articulate the direction from the origin.

Circle in Polar Coordinates

Representing a circle can sometimes be simpler in polar coordinates. A circle centered at the origin with radius \(r\) is described simply by the value of \(r\). It doesn't vary with \(\theta\), as each angle describes a point at a fixed radial distance from the origin. The equation for such a circle is:

• \(r = \text{constant} \), such as \(r = 5 \).

This formula captures all points equidistant from the origin, making it efficient to express and analyze circular shapes. When combined with specific angle bounds, it can define portions of the circle about a polar axis, such as sectors or semicircles limited to certain quadrants.

Angles in Polar Coordinates

Angles are a core component of polar coordinates, defining the orientation of a point. They are measured from the positive x-axis and can extend in either direction. In practical use:

• Angles are usually measured in radians.
• Standard angle ranges include 0 to 2\(\pi\), but can also be expressed in terms of specific quadrants.
• A complete rotation is 2\(\pi\), making polar coordinates ideal for periodic and circular phenomena.

Understanding these angles is crucial, as they define which portion of the circle or plane certain points reside in. For example, specifying an angle range from \(\pi\) to \(\frac{3\pi}{2}\) directs attention to that sector of the circle, targeting points both above and below the x-axis in the specified quadrant.