Problem 40 - College Trigonometry

Question

In Exercises \(31-40\), sketch the region in the \(x y\) -plane described by the given set. $$ \left\\{(r, \theta) \mid 0 \leq r \leq 2 \sin (2 \theta), 0 \leq \theta \leq \frac{\pi}{12}\right\\} \cup\left\\{(r, \theta) \mid 0 \leq r \leq 1, \frac{\pi}{12} \leq \theta \leq \frac{\pi}{4}\right\\} $$

Step-by-Step Solution

Answer

The region is a combination of a limaçon section and a circle sector.

Step 1: Understand the Polar Region

The problem describes a region in polar coordinates. We have two sets of conditions to consider. In polar coordinates, each point is defined by \( (r, \theta) \). The first set of conditions is \( 0 \leq r \leq 2 \sin(2\theta) \) with \( 0 \leq \theta \leq \frac{\pi}{12} \). The second set is \( 0 \leq r \leq 1 \) with \( \frac{\pi}{12} \leq \theta \leq \frac{\pi}{4} \).

Step 2: Sketch the First Region

For the first region, plot the curve \( r = 2\sin(2\theta) \), which is a limaçon, for the range \( 0 \leq \theta \leq \frac{\pi}{12} \). Here, start by calculating \( r \) for \( \theta = 0 \) (where \( r = 0 \)) and for \( \theta = \frac{\pi}{12} \) (where \( r = 2 \sin\left(\frac{\pi}{6}\right) = 1 \)). The curve is symmetrical with respect to the origin because \( \sin \) is symmetric.

Step 3: Sketch the Second Region

For the second region, \( r = 1 \) is a circle with a radius of 1. You should plot a radial line from the origin at an angle \( \frac{\pi}{12} \) and another at \( \frac{\pi}{4} \). The region will be between these two lines and can extend from \( r = 0 \) to \( r = 1 \). This forms a sector of the circle.

Step 4: Union of Both Regions

The sketch of the complete region is the union of the two parts you've drawn. Shade the area for the first region between \( 0 \leq \theta \leq \frac{\pi}{12} \) up to the curve \( r = 2\sin(2\theta) \). Next, shade the area for the second region between the lines \( \theta = \frac{\pi}{12} \) to \( \theta = \frac{\pi}{4} \), within the circle of radius 1. This combined shaded area represents the full region described.

Key Concepts

Limaçon

In polar coordinates, a limaçon is a curve that can take on various shapes, including loops and cardiod forms. The general form for a limaçon is given by \[ r = a + b \cos(\theta) \quad \text{or} \quad r = a + b \sin(\theta) \]depending on the orientation you desire. For this exercise, we focus on \[ r = 2 \sin(2\theta) \],which is a specific type of limaçon.

To plot this curve, calculate the radius, \( r \), at various angles, \( \theta \), within the given range \( 0 \leq \theta \leq \frac{\pi}{12} \). Start with \( \theta = 0 \) to find \( r = 0 \), and proceed to \( \theta = \frac{\pi}{12} \) to calculate \( r = 1 \) as given by \( 2 \sin\left(\frac{\pi}{6}\right) \). The limaçon here is symmetric with respect to the origin due to the properties of the sine function.

Union of Regions

In this exercise, you are given two distinct regions defined by their respective conditions in polar coordinates. The task is to form a single area by forming the union of both regions.

- **First Region**: Determined by \( 0 \leq r \leq 2 \sin(2\theta) \) and \( 0 \leq \theta \leq \frac{\pi}{12} \).- **Second Region**: Encompasses values where \( 0 \leq r \leq 1 \) and \( \frac{\pi}{12} \leq \theta \leq \frac{\pi}{4} \).
To form the union, sketch the first region as a section of the limaçon and the second as a sector of a circle. Visualize both areas, and blend them into one complete sketch by shading the overlapping and connecting sections to represent the union. The union reflects the convergence of these regions into one single domain from the polar perspective.

Polar Curves

Polar curves are a fantastic way to look at shapes and graphs differently than in your standard Cartesian coordinate system \((x, y)\). With polar coordinates, each point on the plane is determined by a distance from the origin and an angle from the positive x-axis. This is symbolically represented as \((r, \theta)\), where \(r\) is the radius and \(\theta\) is the angle in radians.

- **Applications**: Useful for representing spirals, circles, and various symmetrical patterns like roses and limaçons.- **Plotting**: Decide on values of \(\theta\) first, then calculate \(r\). Plot each point and join them to form curves.
Polar curves, in combination with the concept of the union of regions, allow for creative and sophisticated designs, allowing matematicians and scientists to visualize solutions through varying perspectives, enhancing both spatial reasoning and problem-solving skills.