Problem 36 - College Trigonometry

Question

In Exercises \(31-40\), sketch the region in the \(x y\) -plane described by the given set. $$ \left\\{(r, \theta) \mid 1 \leq r \leq 1-2 \cos (\theta), \frac{\pi}{2} \leq \theta \leq \frac{3 \pi}{2}\right\\} $$

Step-by-Step Solution

Answer

The region is between the circle \(r = 1\) and the limaçon \(r = 1 + 2\cos(\theta)\) in the third quadrant.

Step 1: Identify the Polar Inequality

First, recognize the form of the set, \{(r, \theta) \mid 1 \leq r \leq 1-2\cos (\theta), \frac{\pi}{2} \leq \theta \leq \frac{3\pi}{2}\}. This indicates bounds on the radius \(r\) and the angle \(\theta\). Specifically, \(r\) varies between 1 and another function of \(\theta\), while \(\theta\) is between \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\).

Step 2: Convert Polar Function

The inequality \(r \leq 1-2\cos(\theta)\) can describe a polar equation. Rewriting \(r - 1 = -2\cos(\theta)\) lets us manipulate the equation, \(r = 1 + 2\cos(\theta)\), which represents a limaçon of the form \(r = a + b\cos\theta\). It's characteristic because it can create either an inner loop, dimpled, or a cardiod based on values of \(a\) and \(b\).

Step 3: Examine Graphical Characteristics

The given inequality \(1 \leq r \leq 1-2\cos(\theta)\) suggests we are interested in the area in between the circle \(r = 1\) and part of the limaçon \(r = 1 + 2\cos(\theta)\). Since \(\frac{\pi}{2} \leq \theta \leq \frac{3\pi}{2}\), we are considering the left half of the \(xy\)-plane, where \(\cos(\theta)\) is negative, causing \(r\) to increase as \(\theta\) enters \(\pi\) radians (or 180 degrees).

Step 4: Sketch Polar Bounds

Draw the circle \(r = 1\) centered at the origin shows where \(r\) is constantly 1. Next, overlay the limaçon \(r = 1 + 2\cos(\theta)\), making sure to limit it to \(\frac{\pi}{2} \leq \theta \leq \frac{3\pi}{2}\). This limaçon will create a dimpled shape shifting upwards along the negative y-axis due to the range \(\theta\) specified.

Step 5: Indicate Region of Interest

Highlight the area within the boundaries defined by \(1 \leq r \leq 1-2\cos(\theta)\). This essentially colors in the portion between the circle and limaçon curves in the third quadrant, where \(\theta\) extends from \(\frac{\pi}{2}\) to \(\frac{3\pi}{2}\), capturing space between the circle's outer edge and the inside boundary marked by the limaçon.

Key Concepts

Limaçon

A limaçon is a special type of graph that you can plot using polar coordinates. In its basic form, the equation for a limaçon is written as \(r = a + b\cos(\theta)\) or \(r = a + b\sin(\theta)\). These equations create curves that resemble a snail, which is actually where the name 'limaçon' (from French, meaning 'snail') comes from.

• If \( |a| = |b| \), the limaçon forms a cardioid, a heart-shaped figure.
• If \( |a| < |b| \), it produces a limaçon with an inner loop.
• If \( |a| > |b| \), it results in a dimpled or oval-shaped limaçon.

In the problem, the limaçon described by \(r = 1 + 2\cos(\theta)\) doesn't form an inner loop because \(|1| < |2|\). Instead, it displays a dimpled indentation. Understanding these features is crucial as they directly influence how the region is defined and sketched.

Polar Equations

Polar equations describe the relationship between a point's distance from the origin (the "pole") and its angle from the positive x-axis (\(\theta\)). These are different from Cartesian equations that use x and y coordinates.
Using polar equations can make it easier to represent curves that are circular or have symmetrical properties.

• The radius \(r\) in a polar equation can vary depending on the angle \(\theta\).
• In plotting, each point is defined by \((r, \theta)\).
• By varying \(\theta\) and calculating \(r\), you trace out a curve in the polar coordinate plane.

For the given problem, you have the equation \(r = 1 + 2\cos(\theta)\). By finding specific points of \(r\) using \(\theta\) values between \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\), you can accurately sketch the curve. This spans the left-half of the plane often simplifying the depiction of graphs less symmetrically in the Cartesian view.

Trigonometric Inequalities

Trigonometric inequalities involve expressions using trigonometric functions that are set within inequality constraints. They help define specific regions or portions of graphs.
In the exercise, the inequality \(1 \leq r \leq 1-2\cos(\theta)\) defines a set region in the plane.

• \(1 \leq r\): Ensures that the radius is never less than 1, linked to the circle \(r = 1\).
• \(r \leq 1-2\cos(\theta)\): Places an upper bound on the radius where it doesn't exceed the limaçon.

By understanding these inequalities, you focus only on the region of interest, which is the area that lies between the circle and part of the limaçon curve and within the specific angles \(\frac{\pi}{2}\) to \(\frac{3\pi}{2}\). This effectively outlines the portion of the graph to pay attention to when sketching or analyzing any problems using polar coordinates.