Problem 10 - Precalculus with Limits

Question

In Exercises 7-12, identify the type of polar graph. \(r^2=16 \cos 2\theta\)

Step-by-Step Solution

Answer

The polar graph of the equation \(r^2=16 \cos 2\theta\) is a type of lemniscate of size 16.

Step 1: Rewrite the equation to standard form

Notice that the equation can be rearranged to \(r = \sqrt{16\cos 2\theta}\). This equation fits the standard form of lemniscate equations, which is \(r = \sqrt{a\cos 2\theta}\) or \(r = \sqrt{a\sin 2\theta}\), where \(a\) is positive constant. Therefore we can conclude that the given polar graph is a lemniscate.

Step 2: Identify the properties

The value of \(a\) determines the size of the lemniscate. Since \(a = 16\) in our equation, this means the lemniscate is larger than if \(a\) were a smaller positive number.

Key Concepts

Lemniscate

A lemniscate is a special type of polar graph. It resembles the shape of a sideways figure eight or an infinity symbol. This curve is particularly interesting because of its symmetrical and balanced design.
The lemniscate has two loops that intersect at the pole (origin). The curve is created by a particular trigonometric relationship, involving either the cosine or sine function. It appears to loop and turn back, creating those characteristic intersections and symmetrical sides.

• Polar Equation: A lemniscate has a polar equation of the form \(r^2 = a^2 \cos(2\theta)\) or \(r^2 = a^2 \sin(2\theta)\).
• Appearance: The shape is like a two-looped ribbon in the polar coordinate system.

In our specific equation \(r^2 = 16 \cos(2\theta)\), the lemniscate is described by \(a = 4\), because 16 equals \(4^2\). This means each loop sprouts a radius of 4 units from the center, forming a larger lemniscate.

Standard Form

The standard form refers to the way a mathematical expression is written. When dealing with specific types of polar graphs such as a lemniscate, writing the equation in standard form helps to easily identify its type and characteristics.
This is crucial because the standard form reveals the graph structure through constants and function types. In polar coordinates, standard form equations for lemniscates are either \(r = \sqrt{a \cos 2\theta}\) or \(r = \sqrt{a \sin 2\theta}\). These forms directly link the trigonometric value and the `a` value, which determine the size and orientation of the graph.

• Example Form: For our problem, simplifying \(r^2 = 16 \cos 2\theta\) to \(r = \sqrt{16 \cos 2\theta}\) shows its alignment with the standard lemniscate form.
• Why Standard Form?: It's a clear and structured way to analyze different polar graphs, allowing quick conclusions about the graph type and its dimensions.

Utilizing the standard form is an essential skill in understanding and sketching complex polar graphs.

Cosine Function

The cosine function is one of the basic trigonometric functions, widely recognized for its role in describing wave patterns and circular motion. In polar coordinates, the cosine function determines how the graph behaves in terms of symmetry and orientation.
For lemniscates formed with cosine, the equation \(r^2 = a^2 \cos(2\theta)\) manipulates the curves on the horizontal axis due to cosine's nature. It means each loop of the lemniscate is centered around the x-axis, given how cosine values distribute between positive and negative through its cycle.

• Symmetry: Graphs involving cosine in polar coordinates usually display symmetrical properties around the horizontal axis.
• Graph Behavior: Altering the constant \(a\) will stretch or shrink the two loops of the lemniscate along the x-axis.

Understanding the cosine function’s impact is crucial, especially as it frames how polar graphs like the lemniscate develop around particular axes.