Problem 28 - Algebra and Trigonometry

Question

Describe a viewing rectangle, or window, such as [-30, 30, 3] by [-8, 4, 1], that shows a complete graph of each polar equation and minimizes unused portions of the screen. $$ r=\frac{2}{3+3 \sin \theta} $$

Step-by-Step Solution

Answer

This exercise does not have a numerical answer, but the complete graph of \(r=\frac{2}{3+3 \sin \theta}\) can be obtained by converting it to rectangular coordinates and graphing within the window specified. Adjust the scales as needed to minimize unused portions of the screen.

Step 1: Convert Polar to Rectangular Coordinates

First, let's convert the given polar equation to rectangular coordinates for easier graphing. Using the conversion equations \(x = r \cos \theta\), and \(y = r \sin \theta\), where \(r= \frac{2}{3+3 \sin \theta}\). Substituting \(r\) into the equations we get \(x = \frac{2 \cos \theta}{3+3 \sin \theta}\) and \(y = \frac{2 \sin \theta}{3+3 \sin \theta}\) respectively.

Step 2: Determine the Viewing Window

The given window describes in which ranges \(x\) and \(y\) need to be in order to visualize the entire graph correctly. In this case, our \(x\) range is [-30, 30] with a scale of 3, and our \(y\) range is [-8, 4] with a scale of 1. This means our values for \(x\) go from -30 to 30 in increments of 3, and our values for \(y\) go from -8 to 4 in increments of 1.

Step 3: Graph the Rectangular Equations

Now graph the rectangular equations determined in step 1 within the window determined in step 2. There should be one lope at x=0 and another one at every value 360/N for N is 1, 2, 3... etc.

Step 4: Analyze the Graphs in the Viewing Window

Analyze the graphs you've plotted in the viewing window. Identify where they are not seen clearly due to the extreme values of \(x\) or \(y\), and then adjust the scales of the \(x\) or \(y\) axis as needed to minimize unused portions of the screen.

Key Concepts

Rectangular Coordinates

When working with graphs, rectangular coordinates (also known as Cartesian coordinates) are commonly used to plot points on a plane. They are defined as a pair \(x, y\), where \(x\) represents the horizontal position and \(y\) the vertical position of a point. This system is familiar and helps us visualize complex equations more intuitively.

To convert a polar equation like \( r = \frac{2}{3+3 \sin \theta} \) into rectangular coordinates, use the relationships \(x = r \cos \theta\) and \(y = r \sin \theta\). This helps translate polar graphs into the more common rectangular format, making them easier to plot using standard graphing tools.

In this example, substituting the expression for \(r\) gives us:

• \(x = \frac{2 \cos \theta}{3+3 \sin \theta}\)
• \(y = \frac{2 \sin \theta}{3+3 \sin \theta}\)

With these equations, you can graph the function within a viewing window tailored to suit your needs.

Viewing Window

A viewing window is a range of values for \(x\) and \(y\) that defines how an equation is visualized on a graphing calculator or software. It affects how much of the graph you will see and helps in minimizing blank spaces or unnecessary details.

In our exercise, the viewing window is described as \([-30, 30, 3]\) by \([-8, 4, 1]\). This means:

• \(x\) values range from -30 to 30, with a step size of 3.
• \(y\) values range from -8 to 4, with a step size of 1.

Choosing an appropriate window ensures that the main features of the graph are visible and unnecessary parts are minimized, providing a clear and efficient visualization of the equation.

Graphing Equations

Graphing equations, whether in polar or rectangular form, involves plotting the points over a predefined range. This allows you to visualize the data and understand the behavior of the equation.

Start by rewriting the polar equation in rectangular coordinates. Once converted, use the described viewing window to define where your graph will appear. Ensure the key parts of the graph are captured by adjusting \(x\) and \(y\) scales if needed. For \( r = \frac{2}{3+3 \sin \theta} \), plotting within the specified window will reveal its shape correctly.

Remember:

• Check for accuracy of data points.
• Evaluate if the graph's features are fully displayed within the window.
• Make adjustments to center important parts of the graph.

Coordinate Conversion

Converting between polar and rectangular coordinates is a vital skill in mathematics, as it allows flexibility in how we represent and analyze equations.

The basic conversions include:

• From polar to rectangular: \(x = r \cos \theta\) and \(y = r \sin \theta\).
• From rectangular to polar: \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}(\frac{y}{x})\).

This exchange facilitates graphing and interpreting equations in diverse contexts.

During our exercise, the conversion of \(r = \frac{2}{3+3 \sin \theta}\) into \(x\) and \(y\) coordinates demonstrated how such equations could be translated for graphical interpretations. This particular conversion allows us to visualize a polar equation within a rectangular system seamlessly.