Problem 25 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use symmetry to sketch the graph of the polar equation. Use a graphing utility to verify your graph. $$r=3 \sin \theta$$

Step-by-Step Solution

Answer

The graph of this equation is a circle with a radius of 3, centered at the origin. It is symmetric with respect to the x-axis.

Step 1: Understanding the polar equation

The given polar equation is \(r = 3\sin \theta\). This is a standard form of a circle in polar coordinates. The sine function returns a range of values between -1 and 1 and therefore, r which is equal to 3 times the sine of theta can take any values from -3 to 3. The fact that the function is \(\sin \theta\) would suggest symptoms of symmetry.

Step 2: Identifying the symmetry

In polar coordinates, we can identify symmetry about the x-axis when for every point on the plot, its reflection over x-axis is also on the plot. Considering the periodicity and nature of sine function, for every theta, the points (-r, theta) and (r, theta + pi) should exist, which means the graph will be symmetric about the x-axis.

Step 3: Sketching the Graph

Make a table with values of \(\theta\) ranging from 0 to 2\(\pi\). Polar graphs are built based on the radial distance from the origin, you will sketch this table onto polar coordinate which helps you to get a coordinated plot for the poles on the graph.

Step 4: Verifying the Graph with Graphing Utility

Use a graphing calculator or any online graphing tool to input the equation \(r = 3 \sin \theta\) to verify your graph. Such a tool will plot the given function in the polar coordinate system and you can check your drawn graph against this.

Key Concepts

Symmetry in Graphs

When exploring graphs, particularly in polar coordinates, symmetry can be a remarkable property to identify. For the polar equation \(r = 3 \sin \theta\), symmetry is an inherent feature that simplifies the graphing process.

In polar coordinates, symmetry can be examined in relation to the axes or the origin:

• Symmetry over the x-axis implies that if a point \((r, \theta)\) lies on the graph, then \((r, \theta + \pi)\) will also be on the graph.
• You might also encounter symmetry about the y-axis or the origin in other equations, but for \(r = 3 \sin \theta\), the focus is primarily on x-axis symmetry.

Understanding symmetry allows you to predict the behavior of the graph over different angles. This property is especially useful, as it can simplify graphing by reducing the number of calculations needed to plot points correctly. Recognizing this symmetry also supports identifying the graph as a circle in polar coordinates.

Sine Function

The sine function is fundamental in trigonometry and pivotal for polar equations. In the equation \(r = 3 \sin \theta\), the function \(\sin \theta\) directly influences the radius. Let's understand how it operates in this context.

Key characteristics of the sine function relevant to this equation include:

• The sine function oscillates between -1 and 1, meaning \(r\) ranges from -3 to 3 after being multiplied by 3.
• It is a periodic function with a cycle repeating every \(2\pi\), which implies that the graph repeats its pattern after each full rotation.

Understanding these qualities is crucial in plotting the polar graph of \(r = 3 \sin \theta\). Due to the periodic nature, only a section of the graph needs deep analysis because the remainder follows the established pattern. Moreover, the graph reflects the sine curve's smooth, wave-like pattern translated into the polar coordinate system.

Graphing Utilities

Graphing utilities can be a great ally when dealing with complex graphs like those in polar coordinates. The technology available today makes plotting graphs a less daunting task.

Using a graphing utility involves a few straightforward steps:

• Input the equation \(r = 3 \sin \theta\) into the graphing calculator or an online tool.
• Verify your hand-drawn graph by comparing it with the plotted graph on the utility.

Graphing utilities can plot every minute change in angle and radius, thus providing a highly accurate representation of the graph. This is especially helpful in educational settings, allowing students to cross-check their manual graphs against professional-grade ones. Emphasizing the graph's appearance and verifying symmetry can greatly enhance understanding.