Problem 45 - 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 top half of the cardioid \(r=3-3 \cos (\theta)\)

Step-by-Step Solution

Answer

The set-builder notation is \( \{ (r, \theta) \mid 0 \leq r \leq 3 - 3\cos(\theta), 0 \leq \theta \leq \pi \} \).

Step 1: Identify the Cardioid

The given cardioid equation is \(r = 3 - 3\cos(\theta)\). This defines a cardioid in polar coordinates, which is symmetric about the polar axis (horizontal line through the origin).

Step 2: Determine Half of the Cardioid

We are only interested in the top half of the cardioid. This corresponds to the interval \(0 \leq \theta \leq \pi\), which encompasses angles from the positive \(x\)-axis to the negative \(x\)-axis through the upper half-plane.

Step 3: Impose Radial Boundaries

For each angle \(\theta\) within the specified interval, the radial distance \(r\) ranges from 0 (the origin) to the boundary of the cardioid, which is \(3 - 3\cos(\theta)\).

Step 4: Use Set-Builder Notation

The set-builder notation to describe the region is formulated as: \[ \{ (r, \theta) \mid 0 \leq r \leq 3 - 3\cos(\theta), 0 \leq \theta \leq \pi \} \] This notation indicates the collection of all points in the specified region, where the radius \(r\) and angle \(\theta\) satisfy the given conditions.

Key Concepts

Set-Builder Notation

Set-builder notation is a way to describe a set by specifying the properties that its members must satisfy. It's very useful when working with complex mathematical objects or conditions. In our context, it helps express the polar region effectively.
For example, in the problem at hand, we use set-builder notation to characterize the region inside the top half of a cardioid given by the equation \(r = 3 - 3 \cos(\theta)\). Here’s how it looks:

• The notation \(\{ (r, \theta) \mid 0 \leq r \leq 3 - 3 \cos(\theta), 0 \leq \theta \leq \pi \}\) describes every point \((r, \theta)\) which lies in this region.
• The vertical bar "|" stands for "such that." So it reads as "the set of points \((r, \theta)\) such that..."
• The conditions \(0 \leq r \leq 3 - 3 \cos(\theta)\) and \(0 \leq \theta \leq \pi\) ensure we only include the points fitting the top half of the cardioid.

Set-builder notation is concise yet comprehensive, allowing mathematicians to convey precise and complex sets in a clear, readable format.

Cardioid

A cardioid is a type of curve that resembles a heart shape. In polar coordinates, it is often expressed with equations involving trigonometric functions.
The cardioid evaluated here is given by the equation \(r = 3 - 3 \cos(\theta)\). This expression defines a curve which is perfectly symmetrical about the horizontal line (polar axis) passing through the origin.

• The cardioid intersects the origin and reaches a maximum radius along the positive and negative x-axis.
• To analyze its structure, consider that as \(\theta\) varies from 0 to \(\pi\), you trace the upper half of this shape, which is what the original exercise focuses on.
• Polar coordinates make it easier to represent curves like cardioids, since they can be inherently circular in geometry.

This cardioid, thanks to its straightforward trigonometric function, is easy to visualize and important for understanding complex polar regions and their boundaries.

Trigonometric Functions

Trigonometric functions are at the core of many mathematical expressions, especially in describing periodic phenomena. In polar coordinates, these functions help define relationships between angles and radial distances.
In our problem, the function \(- \cos(\theta)\) plays a crucial role in shaping the cardioid's boundary. Here's how:

• The cosine function oscillates between -1 and 1 as \(\theta\) changes. This affects the value of \(-3 \cos(\theta)\), which in turn adjusts the radius \(r\) of the cardioid.
• At \(\theta = 0\), \(\cos(0) = 1\), so \(r = 3 - 3\times1 = 0\). This is why the cardioid intersects the origin at this angle.
• When \(\theta = \pi\), \(\cos(\pi) = -1\), giving \(r = 3 - 3\times(-1) = 6\), the maximum radius in this direction.

Understanding how trigonometric functions integrate into these equations helps demystify their application in describing curves like cardioids and advances comprehension of polar coordinate systems.