Problem 44 - Calculus Early Transcendentals

Question

\(29-48\) Sketch the curve with the given polar equation. $$r^{2}=\cos 4 \theta$$

Step-by-Step Solution

Answer

The curve is an 8-petal rose graph.

Step 1: Understand the Polar Equation

The given polar equation is \(r^2 = \cos 4\theta\). Polar equations relate the radius \(r\) from the origin to a point and the angle \(\theta\) from the positive x-axis. This equation involves trigonometric functions, indicating that the graph will involve symmetry and periodicity.

Step 2: Identify the Symmetry and Periodicity

In polar coordinates, symmetry contributes to the form of the graph. Since \(\cos 4\theta\) has a periodicity of \(\pi/2\), the graph will repeat after every \(\pi/2\) radians. Additionally, because the cosine function is symmetrical about the y-axis, the graph will exhibit symmetry in multiple directions. This suggests a multi-petal form.

Step 3: Determine the Points

To sketch the curve, we can calculate specific points by choosing values for \(\theta\) and solving for \(r\). For example, when \(\theta = 0\), \(r^2 = \cos 0 = 1\), so \(r = 1\). For \(\theta = \pi/8\), \(r^2 = \cos(\pi/2) = 0\), thus \(r = 0\). Continue this process for incremental values of \(\theta\) within the period \([0, \pi/2]\) to understand the behavior.

Step 4: Consider the Effect of Squaring

Since the equation is \(r^2 = \cos 4\theta\), the squaring allows both positive and negative values of \(r\) to satisfy the equation, resulting in points at \(r\) and \(-r\). This leads to the complete 8-petal form because each value of \(4\theta\) repeats four times within \(2\pi\).

Step 5: Sketch the Graph

Using the identified periodicity and symmetry, plot the points you calculated. Due to symmetry in the cosine function and its transformations, a more refined sketch maps out an 8-petal rose with petals aligned symmetrically around the polar axis at regular intervals.

Key Concepts

Polar Equations

Polar equations are a way of expressing mathematical relationships in the polar coordinate system, where each point in the plane is determined by a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). The given equation, \(r^2 = \cos 4\theta\), is a polar equation. Here, \(r\) is the radial distance from the origin, and \(\theta\) is the angle from the positive x-axis. These types of equations are particularly useful for drawing curves that have symmetries and repeating patterns. Unlike Cartesian equations, polar equations are often naturally suited for describing circular and angular relationships.
By understanding polar equations, students can tackle problems that involve rotational symmetry or periodicity in a more intuitive manner. When dealing with such equations, it's essential to know how \(r\) and \(\theta\) vary relative to each other and how these variations affect the overall shape of the graph.

Trigonometric Functions

Trigonometric functions play a crucial role in polar equations, especially functions like cosine (\(\cos\)) and sine (\(\sin\)) that govern the angular component. In polar equations like \(r^2 = \cos 4\theta\), the cosine function affects the values of \(r\) depending on the angle \(\theta\).
The cosine function is known for its periodic and symmetric properties:

• It repeats its values in regular intervals, making it periodic.
• It's symmetric around specific axes, which can influence the symmetry of polar plots.
• In this equation, the polar graph derives part of its structure from the cosine's regular return to zero value points.

Understanding the behavior of trigonometric functions is vital because they provide insight into how the curves behave at various angles, which helps in sketching the graph accurately.

Symmetry and Periodicity

Symmetry and periodicity are fundamental concepts when analyzing graphs in polar coordinates. They help predict the overall structure and repetition patterns of the graph. In the equation \(r^2 = \cos 4\theta\), symmetry and periodicity play key roles.
The periodicity of \(\cos 4\theta\) is due to the fact that the function repeats every \(\pi/2\) radians. This repetition means you only need to understand the behavior within one period to understand the entire graph. Symmetry occurs because the cosine function is inherently symmetric about the vertical axis. For \(r^2 = \cos 4\theta\) on a polar graph:

• The graph repeats after every \(\pi/2\) radians.
• The axis of symmetry further determines that similar shapes appear at regular intervals. Here, this results in an 8-petal rose shape.

Recognizing these patterns can simplify graphing tasks and aid in visualizing curves represented by polar equations.

Graph Sketching

Graph sketching in polar coordinates involves visualizing how a curve extends around the origin based on the polar equation. To sketch the curve of \(r^2 = \cos 4\theta\), consider both the calculated points and the symmetrical patterns. Begin by evaluating \(r\) at various \(\theta\) values spanning one period, \([0, \pi/2]\).
For each \(\theta\), solve for \(r\) to determine where the graph will cross:

• At \(\theta = 0\), \(r = 1\); at \(\theta = \pi/8\), \(r = 0\); and so forth.
• The full period of \(\theta = 0\) to \(2\pi\) will show the entire shape.

Squaring allows for both positive and negative values, resulting in radiating petals at regular intervals. Consider using these properties when plotting because:

• Graphing points from one portion of the period can predict other parts due to symmetry.
• Visualizing the interactions between \(r\) and \(\theta\) helps depict polar roses and other complex curves.

With symmetry in mind, a complete sketch can steadily unfold into an even and beautiful pattern, such as an 8-petal rose, capturing the elegance of polar graphing.