Problem 46 - Algebra and Trigonometry

Question

Use a graphing device to graph the polar equation. Choose the domain of u to make sure you produce the entire graph. $$ r=\sqrt{1-0.8 \sin ^{2} \theta} \quad \text { (hippopede) } $$

Step-by-Step Solution

Answer

Graph the polar equation for \( \theta = [0, 2\pi] \) to see the hippopede.

Step 1: Understand the Polar Equation

The given polar equation is \( r = \sqrt{1-0.8 \sin^2 \theta} \). This describes a curve called a hippopede. The equation determines the distance \( r \) from the origin based on the angle \( \theta \).

Step 2: Determine the Domain for \( \theta \)

For polar equations, \( \theta \) typically covers the interval from \( 0 \) to \( 2\pi \) to ensure a complete cycle around the origin. This full range is important for capturing the symmetry and uniqueness of the curve.

Step 3: Identify the Graphing Technology

Select a graphing calculator or software capable of plotting polar coordinates. Make sure it is set to polar mode rather than Cartesian.

Step 4: Input the Polar Equation

Enter the equation \( r = \sqrt{1-0.8 \sin^2 \theta} \) into the graphing technology. Use the predetermined domain of \( \theta = [0, 2\pi] \) to ensure a full graph is produced.

Step 5: Analyze the Graph

Observe the graph produced. It should depict a symmetric shape, known as a hippopede. The graph completes one cycle from \( 0 \) to \( 2\pi \), showing all features of the hippopede.

Key Concepts

Polar Equation

The polar equation given in the exercise is \( r = \sqrt{1-0.8 \sin^2 \theta} \). This specific equation represents a type of curve known as a hippopede. In polar coordinates, the equation dictates how far the point \( r \) is from the origin, depending on the angle \( \theta \). Unlike Cartesian coordinates which use \(x\) and \(y\) values, polar coordinates focus on the concept of radius and angle.
Polar equations are used to represent a wide array of curves, such as circles, spirals, and more intricate patterns like the hippopede. Polar coordinates are particularly useful in situations involving circular paths or systems often found in physics and engineering.
Understanding these curves starts with mastering how to translate the angle \( \theta \) into a point's distance from the origin, as dictated by the equation. In this case, \(r\) changes dynamically with every change in \(\theta\), painting the intriguing shape of the hippopede.

Graphing Technology

Graphing technology is a valuable tool for visualizing polar equations such as \( r = \sqrt{1-0.8 \sin^2 \theta} \). Graphing calculators or computer software can make graphing these equations much easier.
Here are some common graphing technologies you can use:

• Graphing calculators (e.g., TI-84, TI-89)
• Mathematics software (e.g., Desmos, GeoGebra)
• Programming environments capable of plotting (e.g., Python’s Matplotlib)

These technologies allow you to switch between Cartesian and polar coordinate modes. It's critical when graphing polar equations that polar mode is selected. Before moving on to graphing, check the manual or help resource of your chosen technology to ensure correct settings.
Make sure you input the equation correctly, as a simple miss of a parenthesis or mode selection can drastically change the graph. Also, ensure the domain of \( \theta \) is set correctly to see the complete curve.

Domain of \( \theta \)

The domain of \( \theta \) plays a crucial role in ensuring the entire curve is plotted. In polar equations like \( r = \sqrt{1-0.8 \sin^2 \theta} \), \( \theta \) typically ranges from \(0\) to \(2\pi\). This range, which corresponds to a full circle, ensures every point in the curve is depicted.
Choosing the correct domain involves understanding the periodicity of the angle \( \theta \). For graphs expected to exhibit symmetry or periodic cycles, a complete domain ensures that these properties are fully displayed. Most graphing technologies will automatically understand this cycle, but it's always best practice to verify.
The interval \([0, 2\pi]\) is a standard choice, covering one complete rotation allowing for observations of both symmetry and single-cycle representation. Ensuring this domain avoids missed features and incomplete graph portrayals.

Curve Symmetry

Curve symmetry in polar equations is an important property that enhances understanding of graph shapes. For the equation \( r = \sqrt{1-0.8 \sin^2 \theta} \), symmetry can be observed through the graphical representation, offering insights beyond numerical analysis.
Symmetric curves display identical halves, typically across the origin or a line like \(\theta = \frac{\pi}{2}\). Symmetry can make complex plots simpler to understand and predict. Recognizing symmetry in polar graphs, such as the hippopede's symmetrical loops, helps in reducing the need for thorough point-by-point plotting.
Identifying symmetry involves visual checks using graphing technology, or mathematically verifying if changes in \(\theta\) result in duplicated \(r\) values, which shows that parts of the curve mirror each other. This property of symmetry not only makes the curves aesthetically pleasing but also reduces computational workload, highlighting efficiency in handling polar equations.