Problem 39 - College Trigonometry

Question

In Exercises \(31-40\), sketch the region in the \(x y\) -plane described by the given set. $$ \left\\{(r, \theta) \mid 0 \leq r \leq 2 \sqrt{3} \sin (\theta), 0 \leq \theta \leq \frac{\pi}{6}\right\\} \cup\left\\{(r, \theta) \mid 0 \leq r \leq 2 \cos (\theta), \frac{\pi}{6} \leq \theta \leq \frac{\pi}{2}\right\\} $$

Step-by-Step Solution

Answer

The region is a combination of an elliptical segment and a circular sector in the Cartesian plane.

Step 1: Understand the Polar Equations

We are given two polar equations: \(0 \leq r \leq 2\sqrt{3} \sin(\theta)\) for \(0 \leq \theta \leq \frac{\pi}{6}\) and \(0 \leq r \leq 2\cos(\theta)\) for \(\frac{\pi}{6} \leq \theta \leq \frac{\pi}{2}\). These equations define the regions in terms of polar coordinates.

Step 2: Analyze the First Region

For \(0 \leq \theta \leq \frac{\pi}{6}\), the inequality \(0 \leq r \leq 2\sqrt{3} \sin(\theta)\) describes a region where \(r\) is limited by a sinusoidal curve. The boundary is the curve \(r = 2\sqrt{3}\sin(\theta)\). Convert this to Cartesian using \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\).

Step 3: Convert First Region to Cartesian Coordinates

In Cartesian coordinates, \(r = 2\sqrt{3}\sin(\theta)\) combines with \(y = r\sin(\theta)\) to give \(y = 2\sqrt{3}\sin^2(\theta)\). Use the Pythagorean identity to convert: \(y = 2\sqrt{3} \frac{1 - \cos(2\theta)}{2}\). The curve traces part of an ellipse for \(0 \leq \theta \leq \frac{\pi}{6}\).

Step 4: Analyze the Second Region

For \(\frac{\pi}{6} \leq \theta \leq \frac{\pi}{2}\), \(0 \leq r \leq 2\cos(\theta)\) describes another region. The maximum \(r\) occurs at \(\theta = \frac{\pi}{3}\), making \(r = 1\). This is part of a circular sector. Convert this into Cartesian coordinates.

Step 5: Convert Second Region to Cartesian Coordinates

Convert \(r = 2\cos(\theta)\) to Cartesian: Substitute \(x = r\cos(\theta)\) and \(r^2 = x^2 + y^2\) to get the equation of a circle or ellipse. The boundary is \(x^2 + y^2 = 4\cos^2(\theta)\), tracing part of a circle for \(\frac{\pi}{6} \leq \theta \leq \frac{\pi}{2}\).

Step 6: Sketch the Final Region

Sketch the boundary of both regions on the Cartesian plane. The first region is bounded by the ellipse centered at the origin between \(0 \leq \theta \leq \frac{\pi}{6}\) and includes the corresponding arc. The second region is part of a circular sector from \(\theta = \frac{\pi}{6} \) to \( \theta = \frac{\pi}{2}\). The union of these shapes gives a composite region consisting of part of an ellipse and a circular sector.

Key Concepts

Cartesian Coordinates

Cartesian coordinates provide a simple and familiar way to navigate the plane using perpendicular axes to define points. In this system, each point is specified by an ordered pair \(x, y\), where \(x\) measures horizontal displacement and \(y\) measures vertical displacement.
To convert polar coordinates \(r, \theta\) to Cartesian coordinates, utilize the transformations \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). This conversion is essential when analyzing regions defined in polar equations, allowing them to be understood and graphed within the Cartesian plane.
For example, for the equation \(r = 2\sqrt{3}\sin(\theta)\) within the angle \(0 \leq \theta \leq \frac{\pi}{6}\), it translates in Cartesian as a segment of an ellipse using the identities \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). By substituting and simplifying using trigonometric identities, one can describe these regions more comprehensively.

Polar Equations

Polar equations describe relationships based on distance from a central point and angle from a reference direction. They differ from Cartesian equations mainly because of how they encode points, using \(r\) for radius and \(\theta\) for angle.
The form \(0 \leq r \leq f(\theta)\) implies a radial distance that can vary with angle \(\theta\), effectively tracing curves or regions dependent on how \(f(\theta)\) is formulated. These equations are particularly useful for modeling circular or rotational patterns.
In our given exercise, two key polar equations \(0 \leq r \leq 2\sqrt{3} \sin(\theta)\) and \(0 \leq r \leq 2\cos(\theta)\) define specific sectors between certain angles.
- The first equation outlines a segment that behaves elliptically for a restricted angular domain.- The second pertains to a segment, more circular in nature, allowing easy interpretation of influence from the angle on radius.

Circular Sector

A circular sector is the portion of a circle enclosed by two radii and the arc between them. Imagine a slice of pizza: that's a sector. It's defined by the circle's center, the arc's length, and the angle subtended at the center.
The formula to measure the area of a sector is \((\theta/2\pi) \times \pi r^2 = (\theta/2) r^2\), where \(\theta\) is the central angle in radians and \(r\) is the radius. For instance, look at the region defined by \(\frac{\pi}{6} \leq \theta \leq \frac{\pi}{2}\) with \(0 \leq r \leq 2\cos(\theta)\) as a sector of a circle.
This forces \(r\) to vary, but within a bound tied to the cosine function, ultimately shaping the arc within the sector. In visualization, the circular sector contributes a slice to the overall region by spanning from one angle limit to another, embracing part of the circle's circumference.

Ellipse in Polar Coordinates

Ellipses, when expressed in polar coordinates, can seem less intuitive but offer fascinating geometric interpretations. Typically, an ellipse is characterized as a set of points such that the sum of distances from two foci is constant. However, in polar forms, ellipses can be segments defined by conditions involving \(r\) and \(\theta\).
For example, consider the equation \(r = 2\sqrt{3}\sin(\theta)\), contingent that \(0 \leq \theta \leq \frac{\pi}{6}\). This expression constrains \(r\) to fluctuate with \(\theta\), intuitively indicating a curving path.
Through the transformation \(y = r\sin(\theta)\), the resulting curves trace portions of ellipses on the XY-plane. Recognizing these nuances lets us depict elliptical frames even within polar confines, offering a dual perspective—both coordinate systems enhance comprehensions of such geometric structures.