Problem 44 - 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 \(r=4 \cos (\theta)\) which lies in Quadrant IV.

Step-by-Step Solution

Answer

\(\{ (r, \theta) \mid r = 4 \cos(\theta), -\frac{\pi}{2} \leq \theta \leq 0 \}\)

Step 1: Analyze the Polar Equation

The given polar equation is the circle \(r = 4 \cos(\theta)\). This equation describes a circle with a radius of 2 units centered at (2, 0) in the Cartesian coordinate plane. This type of circle always intersects the origin.

Step 2: Identify the Quadrant IV Region

Quadrant IV is the section of the polar plane where the angle \(\theta\) is between \(-\frac{\pi}{2}\) and 0. Therefore, we need to consider the part of the circle described by \(r = 4 \cos(\theta)\) within this angular range.

Step 3: Define the Region in Set-Builder Notation

To express the region in set-builder notation, we consider both the range of \(\theta\) and the equation of the circle. The set-builder notation combines these requirements to define the region as \(\{ (r, \theta) \mid r = 4 \cos(\theta), -\frac{\pi}{2} \leq \theta \leq 0 \}\). This notation indicates all points \((r, \theta)\) that satisfy the circle equation within the specified angle range.

Key Concepts

Understanding Set-Builder Notation

Set-builder notation is a way to describe a set by specifying the properties that its members must satisfy. In the context of polar coordinates, it spells out the specific conditions for a region. Consider it like a set of instructions that tells you exactly which points belong inside a given region.

In general, set-builder notation is written as \( \{ x \mid \text{condition} \} \), which translates to "the set of all \( x \) such that this condition holds."

For example, in our exercise, the set-builder notation is \( \{ (r, \theta) \mid r = 4 \cos(\theta), -\frac{\pi}{2} \leq \theta \leq 0 \} \). This means we are considering all points \((r, \theta)\) in the polar coordinate plane where:

• The radius \(r\) equals \(4 \cos(\theta)\).
• The angle \(\theta\) is between \(-\frac{\pi}{2}\) and 0, aligning with Quadrant IV.

This approach allows for a concise representation of potentially complex regions by focusing on both the radial and angular components.

Exploring Polar Equations

Polar equations are mathematical expressions that define relationships between the radius \(r\) and the angle \(\theta\) in a polar coordinate system. They are different from Cartesian equations because they focus on points using angles and distances from the origin.

In our exercise, the polar equation \(r = 4 \cos(\theta)\) creates a circle. This circle's center isn't at the origin, unlike in Cartesian coordinates, but shifted towards one side. Converting this equation to Cartesian form results in \[(x-2)^2 + y^2 = 2^2,\] showing a circle centered at \((2, 0)\) with a radius of 2.

To graph polar equations, it can be helpful to calculate values of \(r\) for specific \(\theta\) angles, then plot these in the polar grid. The equation's form tells us how \(r\) changes with \(\theta\), creating intricate shapes like circles, spirals, or even roses. Understanding this helps distinguish which sections of the graph fall within certain angular ranges.

Quadrants in Polar Coordinates

In the polar coordinate system, like the Cartesian plane, the plane is divided into sections called quadrants based on the angle \(\theta\). Each quadrant corresponds to a 90-degree segment of the polar plane.

• Quadrant I: \(0 < \theta < \frac{\pi}{2}\)
• Quadrant II: \(\frac{\pi}{2} < \theta < \pi\)
• Quadrant III: \(\pi < \theta < \frac{3\pi}{2}\)
• Quadrant IV: \(-\frac{\pi}{2} < \theta < 0\)

Quadrant IV, as used in the exercise, involves angles where \(\theta\) is negative, which means measuring clockwise from the positive x-axis.

Understanding which quadrant you are working in is crucial for solving polar problems because it provides insight into the angle range \(\theta\) should cover. It also helps when converting between polar and Cartesian systems, as the sign and direction of the axes influence how points get plotted.