Problem 32 - Precalculus

Question

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: Understanding Polar Coordinates

Polar coordinates are a way of describing locations in a plane using an angle and a distance from an origin point (also called the pole). A point in polar coordinates is represented as (r, \(\theta\)), where r is the distance of the point from the origin, and \(\theta\) is the angle from the positive x-axis to the point in the counter-clockwise direction.

Step 2: Plot the Polar Equation

To plot the equation \(r=\frac{2}{2+3 \sin \theta}\), evaluate the given equation for a variety of \(\theta\) values and plot each resultant point. \(\theta\) is repeatedly varied from 0 to \(2\pi\). With most graphing utilities, you can input the equation directly and they will automatically generate the graph.

Step 3: Identify the Graph

After graphing the polar equation, the next step is to identify the type of the graph. Polar coordinates can often create special types of graphs, including spirals, roses, and circles. The graph that is created by the given polar equation is a limaçon. The limaçon is recognized through its distinctive loop shape.

Key Concepts

Polar Equations

Polar equations are mathematical expressions used in polar coordinates, where the position of a point is determined by the radius \(r\) and the angle \(\theta\). Unlike Cartesian coordinates that use (x, y) to define a point, polar coordinates use the pair (r, \(\theta\)). This system is especially useful for problems involving curves and shapes that display radial symmetry.
When working with polar equations, it is often helpful to understand how the radius changes with respect to the angle. For example, the equation \(r = \frac{2}{2 + 3 \sin \theta}\) describes a complex relationship between \(r\) and \(\theta\).
The meaning of \(r\) here is the distance from the origin to any point on the curve for each angle \(\theta\). As \(\theta\) changes, you must compute each \(r\) value to obtain the shape of the graph. This can result in fascinating and intricate patterns depending on the form of the polar equation.

Limaçon Graphs

Limaçon graphs are a particular type of curve in polar coordinates, derived from equations of the form \[r = a + b \sin \theta\] or \[r = a + b \cos \theta\]. They are notable for their distinctive heart-shaped or dimpled loop, which varies based on the constants \(a\) and \(b\).
In the graph of the polar equation \[r = \frac{2}{2+3 \sin \theta}\], the specific arrangement causes a loop characteristic of a limaçon curve. Analyzing the equation reveals that the denominator significantly impacts the curve's shape, tending towards a loop when \(b > a\) in the given form.

• Looped Limaçon: These occur when \(b > a\), resulting in an inner loop as seen in the given equation.
• Dimpled Limaçon: Occur when \(b < a\), yielding a less pronounced curve.

Each type comes with its unique symmetry and visual appeal, predominantly utilized in physics and engineering scenarios to represent fields with rotational symmetry.

Graphing Utilities

Graphing utilities are tools that allow users to visualize equations graphically, which is particularly beneficial when working with complex mathematical expressions like polar equations.
These utilities, which can be software applications or online platforms, allow users to enter polar equations directly, offering an immediate visual representation. For example, inputting \[r = \frac{2}{2 + 3 \sin \theta}\] into a graphing calculator will render its limaçon loop.
Using a graphing utility can save time and prevent potential human error that may occur if plotting manually. It allows for easy exploration of \(\theta\) as it ranges from 0 to \(2\pi\), providing a complete graphical representation with minimal effort. Users can:

• Modify the ranges and scales for better visual comprehension.
• Observe animation over different \(\theta\) values to understand dynamics visually.
• Evaluate specific characteristics, such as symmetry or intercepts.

These advantages make graphing utilities nearly indispensable, especially when exploring the rich and varied world of polar graphs.