Problem 33 - Precalculus Mathematics for Calculus

Question

Sketch a graph of the polar equation. $$r=\sqrt{3}-2 \sin \theta$$

Step-by-Step Solution

Answer

The graph is an inner-loop limaçon with outer and inner loops corresponding to specific \( \theta \) ranges.

Step 1: Understanding the Polar Equation Form

The given polar equation is in the form \( r = a - b \sin \theta \), where \( a = \sqrt{3} \) and \( b = 2 \). This indicates that the graph of the equation is a limaçon.

Step 2: Identify the Limaçon Type

To determine the type of limaçon, compare \( a \) and \( b \). Since \( a = \sqrt{3} \) and \( b = 2 \), and \( \sqrt{3} < 2 \), this represents an inner-loop limaçon because \( a < b \).

Step 3: Find Key Angles and Points

Calculate points for key angles to sketch the graph accurately. For example:- At \( \theta = \frac{\pi}{2} \), \( r = \sqrt{3} - 2(1) = \sqrt{3} - 2 \).- At \( \theta = \frac{3\pi}{2} \), \( r = \sqrt{3} - 2(-1) = \sqrt{3} + 2 \).Use these to plot the graph.

Step 4: Determine the Inner and Outer Loop Ranges

The inner loop occurs when \( r < 0 \). Solve \( \sqrt{3} - 2 \sin \theta < 0 \) to find \( \sin \theta > \frac{\sqrt{3}}{2} \). The range for \( \theta \) is \( \left( \frac{\pi}{3}, \frac{2\pi}{3} \right) \). The remaining angles will generate the outer loop.

Step 5: Sketch the Graph with Critical Features

Begin at \( \theta = 0 \) and visually progress to \( \theta = 2\pi \). Plot points calculated and sketch the inner loop between \( \theta = \frac{\pi}{3} \) to \( \frac{2\pi}{3} \), then complete the limaçon with the outer loop sections. The inner loop will cross through negative \( r \) values and return to positive values.

Key Concepts

Limaçon

Limaçon is a term derived from the French word for snail, and it describes a particular family of curves graphed using polar coordinates. A limaçon can come in various forms depending on the values of specific parameters in its polar equation. In general, the equation of a limaçon can be written as either \( r = a \, \pm \, b \sin \theta \) or \( r = a \, \pm \, b \cos \theta \). When plotting these curves, you'll find that the graph resembles a snail-like shape which can exhibit different features, such as an inner loop.
To determine what the shape will look like, compare the constants \( a \) and \( b \) from the equation. The relationship between \( a \) and \( b \) indicates whether the limaçon will have an inner loop, a dimpled shape, a cardioid, or will behave as a convex shape. In the equation \( r=\sqrt{3}-2 \sin \theta \), \( a < b \), signaling that an inner loop appears due to the value of \( a \) being smaller than \( b \). This characteristic is critical as it influences both the shape and how the graph needs to be plotted within its polar coordinate system.

Graph Sketching

Graph sketching is an essential skill in visualizing functions and their consequences. Sketching a polar graph is slightly different compared to a Cartesian graph, largely due to how points are plotted. In polar graphs, points are given in the form \( (r, \theta) \), where \( r \) is the distance from the origin, and \( \theta \) is the angle from the positive x-axis.
Step by step, a graph is sketched by identifying key points and angles. In the context of a limaçon with an inner loop, like the one described by \( r = \sqrt{3} - 2 \sin \theta \), we identify crucial angles where the curve's principal features occur. These include key transitions from negative to positive values of \( r \) such as \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \).

• Start plotting from \( \theta = 0 \).
• Proceed through angles to \( \theta = 2\pi \).
• Plot key points calculated in the equation, ensuring the inner and outer loops are captured.

The objective is to ensure the shape accurately represents the mathematical properties outlined by the equation, particularly focusing on the characteristics of the limaçon.

Polar Coordinates

Polar coordinates provide a unique way of describing locations in a plane. Unlike Cartesian coordinates, which use \( (x, y) \) pairs, polar coordinates use \( (r, \theta) \) to specify a point's distance and direction from a fixed point called the pole, essentially the origin. This system is particularly useful for graphing equations involving angles and curves more naturally modeled in circular coordinates.
In any polar equation like \( r = f(\theta) \), \( r \) determines how far the point is from the pole, and \( \theta \) indicates the direction or angle. These are especially handy when working with conic sections or curves that inherently involve rotational symmetry, like limaçons or spirals.

• \( r < 0 \) implies the vector points in the opposite direction of \( \theta \).
• As \( \theta \) increases, it represents a counterclockwise movement.
• The full revolution around the circle is completed with \( \theta = 2\pi \).

Switching between polar and Cartesian coordinates requires transformations using trigonometry, for example, \( x = r \cos \theta \) and \( y = r \sin \theta \). Understanding these principles helps in the conversion and understanding of different graphical representations.