Problem 22 - 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. $$ r=\frac{16}{5-3 \cos \theta} $$

Step-by-Step Solution

Answer

The viewing window or rectangle that minimizes unused portions of the screen and shows the complete graph of the given polar equation, \(r=\frac{16}{5-3 \cos \theta}\), can be determined by converting the polar equation into Cartesian coordinates, simplifying them and determining the range. The range [-30, 30, 3] by [-8, 4, 1], forms the required viewing rectangle.

Step 1: Understand the Polar Equation

The given polar equation is \(r=\frac{16}{5-3 \cos \theta}\). The r in the equation represents the radial coordinate or the radius - the direct distance from the origin to the point, and theta (\(\theta\)) is the angular coordinate or angle formed with the positive x-axis.

Step 2: Convert Polar to Cartesian Equation

You can represent the polar coordinates (r, θ) in Cartesian coordinates (x, y) by using two fundamental relations: \(x = r \cdot \cos(\theta)\) and \(y = r \cdot \sin(\theta)\). Substitute \(r = \frac{16}{5-3 \cos \theta}\) into these relations to get x and y in terms of θ.

Step 3: Simplify the obtained Cartesian Equations

Upon simplifying, the Cartesian representation of the polar equation can be obtained. This triggers the appropriate axes along which the graph of the equation can be drawn.

Step 4: Determine the Viewing Window

Use the Cartesian Equations to determine the maximum and minimum values of x and y. This forms the viewing rectangle for the polar graph. The provided window [-30, 30, 3] by [-8, 4, 1], represents the ranges of x and y respectively.

Key Concepts

Polar Equation

When studying mathematics, you'll come across various coordinate systems, each with its unique way of representing points in a plane. Polar equations are such representations, where the position of a point is determined by its distance from a fixed point (usually the origin), and its angle with respect to a fixed direction (typically the positive x-axis). The standard form of a polar equation is expressed as

\( r = f(\theta) \)

where \(r\) is the radial coordinate and \(\theta\) is the angular coordinate. In the exercise, the polar equation given is \(r=\frac{16}{5-3 \cos \theta}\), it defines a unique set of points in the polar coordinate system that will form a certain path or shape when graphed. Understanding polar equations is essential for graphing complex curves that are more naturally represented in polar form rather than Cartesian coordinates.

Cartesian Coordinates Conversion

Converting between polar and Cartesian coordinates is a critical skill in geometry and calculus. The main idea is to translate the polar point \( (r, \theta) \) into its Cartesian form \( (x, y) \) which is more familiar to many students. This is done through the formulas:

\(x = r \cdot \cos(\theta)\)
\(y = r \cdot \sin(\theta)\)

When you take the polar equation \(r=\frac{16}{5-3 \cos \theta}\) and apply these formulas, you get the x and y values in terms of \(\theta\). This conversion is fundamental when graphing polar equations or integrating complex equations over specific areas, as it can simplify calculations and visualization by shifting to the Cartesian plane.

Graphing Polar Equations

Graphing polar equations can often seem challenging, but with understanding and practice, it becomes an intriguing aspect of mathematics. To graph a polar equation, such as \(r=\frac{16}{5-3 \cos \theta}\), you would plot points for various values of \(\theta\) and find the corresponding \(r\) value. These points are plotted on a polar grid where the angle \(\theta\) is measured from the positive x-axis, and \(r\) is the distance from the origin.

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Step-by-Step Graphing\

Start by choosing angle increments (like 10 degrees or, for more precision, smaller increments). Calculate \(r\) for each \(\theta\), then move counterclockwise from the positive x-axis by the angle \(\theta\), going outwards from the origin the distance of \(r\). When each point is plotted, connect them smoothly to form the shape of the graph. The beauty of the polar graph becomes apparent as you plot each point and see the curve taking shape.

Viewing Rectangle

In the realm of graphing on digital platforms, whether it be a calculator screen or computer software, a 'viewing rectangle' is crucial. It represents the part of the graph that you will be able to see on your screen. The provided viewing rectangle is denoted as [-30, 30, 3] by [-8, 4, 1].

Here’s what those numbers mean:

• The first set, [-30, 30, 3], gives the horizontal limits for the graph. The graph's x-values will range from -30 to 30, with a 'scale' marking every 3 units.
• The second set, [-8, 4, 1], outlines the vertical constraints. The y-values will range from -8 to 4, with a scale marking each unit.

Selecting an appropriate viewing rectangle is important to ensure that you can see the complete graph without wasting screen space. To do so, one should assess the behavior of the function being graphed to identify the necessary range for both \(x\) and \(y\) values.