Problem 69 - Precalculus

Question

Graph the following equations: a. \(r^{2} \theta=5 \cos \theta, 0 \leq \theta \leq 2 \pi\) b. \(r^{2} \theta=5 \cos (2 \theta), 0 \leq \theta \leq \pi\) c. \(r^{2} \theta=5 \cos (4 \theta), 0 \leq \theta \leq \frac{\pi}{2}\) What do you notice about all of these graphs? Suppose that the movement of the tip of the sword in a game is governed by these graphs. Describe what happens if you change the domain in (b) and (c) to \(0 \leq \theta \leq 2 \pi\).

Step-by-Step Solution

Answer

All graphs are symmetrical with different numbers of petals; changing the domain in (b) and (c) completes their patterns.

Step 1: Understanding the Polar Equation

The equation is given in polar coordinate form. The general form is \( r^2 \cdot \theta = 5 \cos(k\theta) \), where \( k \) is a parameter that affects the number of petals in the plot. In this exercise, we need to understand the behavior of the graph as \( heta \) progresses through the specified intervals.

Step 2: Graphing equation (a)

For the equation \( r^2 \theta = 5 \cos \theta \) within the domain \( 0 \leq \theta \leq 2\pi \), isolate \( r \) by solving \( r^2 = \frac{5 \cos \theta}{\theta} \). This equation traces a spiral-like curve that loops towards and away from the origin. At \( \theta = 0 \), \( r^2 \) becomes singular, and it varies otherwise, creating a symmetrical shape relative to the polar axis.

Step 3: Graphing equation (b)

For the equation \( r^2 \theta = 5 \cos(2\theta) \) with the domain \( 0 \leq \theta \leq \pi \), similarly, solve for \( r \): \( r^2 = \frac{5 \cos(2 \theta)}{\theta} \). This equation produces a two-petal rose or a lemniscate-like shape, using symmetry on the polar coordinate plane. Extending the domain impacts symmetry and form.

Step 4: Graphing equation (c)

For the equation \( r^2 \theta = 5 \cos(4\theta) \) within \( 0 \leq \theta \leq \frac{\pi}{2} \), solve \( r^2 = \frac{5 \cos(4 \theta)}{\theta} \). This equation results in four petals having symmetry about both axes, appearing denser due to the increased \( k \) value in the cosine function. With the domain confined, only one petal is drawn; extending the domain enlarges the pattern.

Step 5: Notice the Symmetry and Petal Patterns

All graphs exhibit symmetry due to the cosine function, and the parameter \( k \) determines the number of petals: graph (a) has a single petal, graph (b) has two, and graph (c) shows four. Adjusting the domain for graphs (b) and (c) to \( 0 \leq \theta \leq 2\pi \) results in completed symmetrical patterns showing all petals.

Key Concepts

Polar Equations

Polar equations are a way to express mathematical relationships using polar coordinates. In these coordinates, a point is defined by the distance from the origin (radius) and the angle from the positive x-axis. This is different from Cartesian coordinates, where points are defined using x and y coordinates.

Polar equations often involve trigonometric functions such as sine and cosine, which help define curves like spirals and roses on a polar graph. These functions are central when dealing with polar graphs because they create symmetric and often elegant patterns.

In this exercise, we see polar equations in the form \( r^2 \theta = 5 \cos(k\theta) \). The variable \( k \) significantly impacts the graph's complexity and the number of curves, or petals, it creates. Understanding how to manipulate these equations is critical to mastering graphing in polar coordinates.

Petal Patterns

Petal patterns in polar graphs are visually striking and are associated with equations involving trigonometric functions like sine and cosine. These patterns are characterized by multiple loops or 'petals' spread around a central point.

The number of petals a polar equation generates depends on the multiplier \( k \) in the function \( \cos(k\theta) \).

• If \( k \) is an odd integer, the graph will have \( k \) petals.
• If \( k \) is an even integer, the graph will have \( 2k \) petals.

Using the above rules, graph (a) with \( k=1 \) results in a single spiral. Graph (b) shows two petals with \( k=2 \), while graph (c) with \( k=4 \) shows four petals when fully extended in its domain. The petal pattern becomes more complex as \( k \) increases, resulting in intricate designs.

Symmetry in Polar Graphs

Symmetry is a key feature of many polar graphs. It enhances the visual appeal and mathematical properties of graphs.

A crucial aspect of symmetry in polar graphs is understanding how it arises from trigonometric functions like cosine. In this case, cosine functions inherently provide even symmetry.

• Graphs are symmetric about the polar axis (horizontal line running through the origin).
• For equations with even \( k \), symmetry about both the polar axis and the y-axis becomes apparent.

For example, equation (c) presents a symmetrical pattern around both axes due to the term \( \cos(4\theta) \). In extending the domain for graphs (b) and (c) to \( 0 \leq \theta \leq 2\pi \), we observe symmetrical, complete floral-like structures, illustrating the beautiful inherent symmetry of these graphs.

Graphs of Trigonometric Functions

Trigonometric functions play a significant role in shaping the curves in polar graphs. When used in polar equations, they generate various patterns depending on their period and frequency.

For cosine functions, as seen in our example polar equations, the period determines how many times the cosine wave is divided within the complete rotation of \( 2\pi \) radians (a full circle). The frequency, dictated by \( k \) in \( \cos(k\theta) \), affects how close or far apart these oscillations occur.

• Graphs can include symmetric patterns in a rotational manner around the origin.
• They display 'roses' or spiral patterns depending on the specifics of the equation.

For the given exercise, these graphs illustrate transformations of trigonometric functions into geometric art, where varying \( k \) alters the graphical complexity from simple to ornate, showcasing the integral role of trigonometric functions in polar graphing.