Problem 20 - Essential Calculus

Question

In Exercises \(17-20,\) use a graphing utility to graph the polar equation. Identify the graph. \(r=\frac{2}{2+3 \sin \theta}\)

Step-by-Step Solution

Answer

The graph of the polar equation \(r=\frac{2}{2+3 \sin \theta}\) is a limaçon.

Step 1: Convert the Polar Equation to Rectangular Form

Polar equations can be converted to rectangular form by replacing \(r\) with \(\sqrt{x^2+y^2}\), \(sin\theta\) with \(\frac{y}{r}\), and \(cos\theta\) with \(\frac{x}{r}\). However, in this case, it may not be necessary because we are using a graphing utility which accepts polar equations.

Step 2: Input the Polar Equation into the Graphing Utility

To input the polar equation \(r=\frac{2}{2+3 \sin \theta}\) into a graphing calculator or software that accepts polar equations, simply enter the equation as is. We also make sure to use 'θ' as the variable.

Step 3: Observe and Identify the Graph

After graphing the equation, observe the general shape and characteristics of the curve. Then, identify the type of graph. For example, the graph could be a circle, spiral, rose, or lemniscate. For this equation, the graph will be a limaçon, a type of polar graph characterized by loops or dimples.

Key Concepts

Understanding Graphing Utilities

A graphing utility can be an invaluable tool when studying complex mathematical functions, such as those expressed in polar coordinates. These utilities range from physical graphing calculators to software programs designed for computers and mobile devices. They help visualize equations by plotting points and generating a curve that represents the function.

For students, learning how to operate graphing utilities effectively is critical. By inputting a polar equation, such as \(r=\frac{2}{2+3 \sin \theta}\), you can promptly obtain a visual representation of the equation. It's also vital to understand the settings on your graphing utility, ensure that it's set to the correct mode (in this case, polar rather than rectangular), and use the correct variable, commonly 'θ' in polar equations. By mastering these tools, students can more easily interpret the behavior and characteristics of complex equations like those resulting in limaçons.

Polar to Rectangular Conversion

Occasionally, you might need to convert a polar equation to its rectangular form to better understand its properties or to use different mathematical techniques. The polar to rectangular conversion leverages the relationships between the polar coordinates (\(r, \theta\)) and the rectangular coordinates (\(x, y\)).

The key conversions to remember are \(x = r \cos \theta\) and \(y = r \sin \theta\). Inversely, \(r = \sqrt{x^2+y^2}\) and \(\theta = \arctan(\frac{y}{x})\). Applying these to the given equation \(r=\frac{2}{2+3 \sin \theta}\), you might replace \(r\) with \(\sqrt{x^2+y^2}\) and \(\sin \theta\) with \(\frac{y}{\sqrt{x^2+y^2}}\) to convert it. Nevertheless, for graphing purposes, using the polar form directly in a graphing utility optimized for polar equations is often more straightforward and efficient.

Characteristics of Limaçon

In polar graphing, a limaçon is a fascinating and distinctive type of curve. It is defined by an equation of the form \(r = a + b \sin \theta\) or \(r = a + b \cos \theta\), where \(a\) and \(b\) are constants. The shape of a limaçon varies depending on the relationship between these constants. It can be a simple loop, a cardioid (when \(a = b\)), or have an inner loop (if \(b > a\)).

The limaçon described by the equation \(r=\frac{2}{2+3 \sin \theta}\) is the result of such a polar function where, instead of a sum, we have a ratio that governs the radius length for each angle \(\theta\). This creates a graph with a dimple or a loop, which is characteristic of limaçons. Understanding the general behavior of such shapes is beneficial because it helps recognize them when plotting polar equations, enabling a deeper comprehension of polar coordinate graphing and the diverse range of curves it can produce.