Question

In Problems \(46-49\), use a computer or graphing calculator to graph the given equation. Make sure that you choose a sufficiently large interval for the parameter so that the entire curve is drawn. $$ r=\sqrt{1-0.5 \sin ^{2} \theta} $$

Short answer

Graph the equation over \( \theta \) from 0 to \(2\pi\).

Step-by-step solution

Understand the Equation Type

The given equation is in polar form: \( r = \sqrt{1-0.5 \sin^2 \theta} \). Our task is to graph this polar equation using a calculator or graphing software. Here, \( r \) gives us the radial distance from the origin as a function of the angle \( \theta \).

Determine the Suitable Interval for \( \theta \)

In polar graphs, \( \theta \) typically varies between \( 0 \) and \( 2\pi \) radians to capture a full rotation around the circle. This interval should suffice to plot the complete graph of \( r \).

Set Up the Graphing Tool

Access a graphing calculator or appropriate software that can handle polar coordinates. Enter the equation \( r = \sqrt{1-0.5 \sin^2 \theta} \) under the polar coordinate system settings.

Graph the Equation

Input the interval for \( \theta \) from \( 0 \) to \( 2\pi \). Adjust any necessary settings to ensure the graph is smooth and covers the entire specified interval. The graph should appear on your screen with the given settings.

Analyze the Graph

Examine the graph for key features such as symmetry, loops, or any regions where the curve might not be well-defined. Make notes on whether the curve meets expectations based on the function's behavior.

Key concepts

Graphing Calculator

A graphing calculator is an essential tool for visualizing equations, especially those that use polar coordinates like the exercise's equation. Graphing calculators can plot complex functions across multiple dimensions. This capability makes them perfect for understanding how the radial distance and angle in polar coordinates are related. These calculators allow users to input the polar equation directly and view a precise graphical representation of its curve on the polar plane. To get started, ensure that your calculator is set to accept polar inputs, usually found through a mode or function setting. Then, simply enter your equation and select an appropriate angle interval to view the graph. It's worth noting that graphing calculators can also highlight unique features of polar graphs, such as symmetry or hidden loops, which may not be immediately apparent from the equation alone.
Graphing calculators are also essential for checking math assignments, just like the ones where you're asked to graph a polar equation. Make sure to study the visual output, as it will aid in understanding the complex nature of the polar coordinate system.

Radial Distance

Radial distance, often represented by the letter \( r \), signifies how far a point is from the origin in polar coordinates. It’s a crucial part of understanding polar equations because it explains the relationship of a point in a polar space to the angle \( \theta \). In our given equation, \( r = \sqrt{1-0.5 \sin^2 \theta} \), the radial distance is not constant but varies with the angle \( \theta \).
As \( \theta \) changes, \( r \) adapts, resulting in unique curves when plotted. The radial distance in polar graphs can often lead to interesting patterns like spirals, loops, or periodic curves depending on the mathematical form. When visualizing these graphs, radial distance helps provide a more intuitive grasp of "where" in the plane each point lies in relation to the origin. Don't assume a static circle; polar graphs are dynamic and shaped by the way \( r \) interacts with \( \theta \).
Understanding radial distance is also beneficial when interpreting the graphical output from your calculator, as the radial changes can highlight the nature of the equation. Moreover, it helps recognize whether all potential points are drawn by ensuring all relevant \( \theta \) values are covered in the graphing interval.

Angle Interval

The angle interval for \( \theta \) in polar coordinates is pivotal for generating a complete graph of any polar equation. Typically, the interval from \( 0 \) to \( 2\pi \) radians covers one full cycle around the polar plane, but this can vary based on specific function requirements. In the present exercise, this interval ensures we capture the entire graph of \( r = \sqrt{1-0.5 \sin^2 \theta} \).
If the interval is too small, you may miss crucial parts of the graph, possibly resulting in an incomplete or misleading representation. However, using extended intervals may also be beneficial in revealing additional symmetrical patterns or behaviors, especially when the function has periodic properties.
When selecting the angle interval, always match the interval to the mathematical behavior you expect from the function. The standard \( 0 \) to \( 2\pi \) range is often enough, but experimenting beyond this can be insightful depending on the polar equation at hand.

• A comprehensive interval guarantees a complete and accurate visualization.
• It allows students to observe the graph's periodic behavior and symmetry.
• Analyzing different intervals can provide deeper insights into the graph's properties.

Polar Equation

A polar equation, such as \( r = \sqrt{1-0.5 \sin^2 \theta} \), offers a distinct method of representing curves that differ from the Cartesian coordinate system. In polar terms, each curve is a combination of \( r \), the radial distance, and \( \theta \), the angle. These two variables provide a powerful toolset for defining not just circles, but also blossoming flowers, spirals, and more intricate figures.
This particular equation might look complex because of the square root and the sine function, which suggests non-linear characteristics. Sine functions introduce periodic behavior into the equation, leading to intriguing cyclical patterns on the polar plane. Recognizing what type of graph a polar equation will produce comes with practice and intuition.
Studying polar equations provides insights into different mathematical phenomena, and is an excellent bridge to understanding periodicity and symmetry in math. Here are some tips for handling polar equations:

• Always note the type of trigonometric functions involved as they define periodic behavior.
• Understand that polar equations often describe more complex and beautiful curves than Cartesian ones.
• Use technology like graphing calculators to efficiently visualize and comprehend these curves.

By comprehending each component of the equation, such as \( r \) and \( \theta \), one can gain a deep understanding of the geometry of polar coordinates.