Problem 84 - Precalculus: Functions and Graphs

Question

Sketch the graph of the polar equation. $$r=8 \cos 5 \theta$$

Step-by-Step Solution

Answer

The graph is a 5-petaled rose with petals of length 8.

Step 1: Identify the Type of Polar Equation

The given equation is \( r = 8 \cos 5\theta \). This is a polar equation that involves the cosine function and a multiple angle, making it a type of rose curve. The coefficient of \( \theta \) is 5, which affects the number of petals of the rose curve.

Step 2: Determine Number of Petals

Since the equation is \( r = a \cos n\theta \) with \( n = 5 \), the number of petals is equal to \( n \) because \( n \) is odd. Therefore, this polar rose will have 5 petals.

Step 3: Find Length of the Petals

The length of each petal is determined by the coefficient \( a \) of the cosine function. In this equation, \( a = 8 \), so each petal will have a length of 8 units from the pole (origin).

Step 4: Determine Symmetry and Orientation

Polarly, a \( \cos \) rose is symmetric about the polar axis (the horizontal line through the pole). The petals are evenly distributed around the pole, with one petal centered along the polar axis.

Step 5: Sketch the Graph

Begin by sketching a circle centered at the pole. Draw one petal along the positive horizontal axis extending outward to a length of 8 units. Next, place additional petals by evenly dividing the full angle (360° or \( 2\pi \) radians) into 5 equal parts. Each angle between petals is \( \frac{360°}{5} = 72° \) or \( \frac{2\pi}{5} \) radians. Plot each petal symmetrically around the circle.

Step 6: Label and Finalize the Graph

Label each petal's maximum point at radius 8 from the origin, and ensure all petals are spaced equally at 72° or \( \frac{2\pi}{5} \) radians apart. Make sure your graph clearly shows symmetry about the polar axis.

Key Concepts

Polar Equations

Polar equations are a powerful tool for graphing curves on a plane using a system that revolves around a central point, the pole, which is similar to the origin in a Cartesian system. The foundation of a polar equation lies in its ability to represent a location through an angle and a radius. The general form of a polar equation is expressed as \( r = f(\theta) \), where \( r \) is the radius from the pole, and \( \theta \) is the angle measured from a fixed direction.

In the world of polar graphing, you'll often encounter equations involving trigonometric functions like sine and cosine. These equations can create a variety of interesting and complex shapes. By manipulating the functions and variables within these equations, different types of curves can be represented, including circles, spirals, and the striking rose curves.

Understanding the basics of polar equations is crucial, as they offer a unique way to view complex shapes and phenomena that aren't as easily understood with linear equations in rectangular coordinates. They provide a fascinating insight into rotation and symmetry structures in mathematics.

Rose Curves

Rose curves are a beautiful example of patterns that can be created using polar equations. These are typically of the form \( r = a \cos n\theta \) or \( r = a \sin n\theta \). The fascinating aspect of rose curves lies in their petal-like shapes, which depend on the coefficient \( n \).

When \( n \) is an integer:

• If \( n \) is odd, the rose curve will have \( n \) petals.
• If \( n \) is even, the curve will feature \( 2n \) petals.

In our given equation, \( r = 8 \cos 5\theta \), we note that \( n = 5 \), hence the rose curve exhibits 5 petals. The equation drives the symmetry and elegant design, leaving paths in equal angular increments around the circle. Whether through music, nature, or mathematics, the exploration of rose curves truly epitomizes the intersection of beauty and numeric precision.

Symmetry in Polar Coordinates

Symmetry is a crucial aspect of plotting polar graphs, especially when dealing with rose curves. For a polar equation, symmetry tells us how the graph behaves with respect to various axes and also aids in efficiently sketching the graph.

When working with rose curves and trigonometric functions, it's important to identify the symmetry to understand the full visual outcome. In the case of the equation \( r = 8 \cos 5\theta \), it is symmetric about the polar axis—the horizontal axis passing through the origin.

• A rose curve that employs a \( \cos \) function typically manifests symmetry about this polar axis.
• When dealing with \( \sin \), the graph is often symmetric about the vertical line.

Recognizing symmetry helps efficiently place the petals in their correct positions without needing to graph the complete angle range, thus making graphing faster and simpler.

Petal Length in Polar Graphs

The length of petals in polar graphs is an essential characteristic that determines how far each petal stretches from the pole. For rose curves, it is influenced by the coefficient \( a \) in the equation \( r = a \cos n\theta \) or \( r = a \sin n\theta \).

In the specific case of \( r = 8 \cos 5\theta \), the coefficient \( a = 8 \) dictates that each petal reaches a maximum radial distance of 8 units from the pole. This concept is especially crucial for accurately plotting the graph because each petal should extend equally and symmetrically from the center.

Understanding the petal length not only ensures aesthetic precision when sketching but also reinforces comprehension of how the coefficients affect the geometric properties of polar graphs. Accurately plotting the petal length alongside the symmetry of the graph is fundamental in mastering polar graphing techniques.