Problem 30 - Algebra and Trigonometry

Question

15–36 Sketch the graph of the polar equation. $$r=2 \cos 3 \theta \quad \text {(three-leaved rose)}$$

Step-by-Step Solution

Answer

The graph is a three-leaved rose centered at the origin with petals extending to radius 2.

Step 1: Understand the Polar Equation

The given equation is \( r = 2 \cos 3 \theta \). This is a type of polar equation known as a 'rose curve.' The coefficient of \( \theta \), which is 3 in this case, indicates the number of petals the rose will have when \( n \) is odd, barring any transformations to the function.

Step 2: Determine Key Points

Identify important points by choosing specific angles \( \theta \) and calculating \( r \). Since the cosine function is periodic with a period \( 2\pi \), evaluating the function at angles such as \( 0\), \( \frac{\pi}{6} \), \( \frac{\pi}{3} \), \( \frac{\pi}{2} \), and \( \pi \) will help trace the curve. The key values to consider for small rose petals might be the maximum \( r = 2 \), minimum \( r = -2 \), and intersecting points where \( r = 0 \).

Step 3: Sketch the Graph

Plot the points determined in Step 2, connecting them smoothly to form three 'petals.' Since this is a symmetrical graph, it will repeat every \( \frac{2\pi}{3} \). Each petal will touch the outermost circle at a radius of \( r = 2 \) and intersect itself at the origin (where \( r = 0 \)). Keep in mind the symmetry around the origin dictated by the nature of cosine.

Step 4: Finalize the Graph Appearance

Ensure that the petals are evenly spaced around the origin and have the correct amplitude of 2, extending in both positive and negative directions. Since the function is \( 2 \cos 3\theta \), the symmetry and the periodicity ensure that you only need to graph part of the curve from \( 0 \) to \( 2 \pi \), and the pattern will naturally extend.

Key Concepts

Rose Curve

A rose curve is a fascinating type of polar graph represented by equations of the form \( r = a \cos(n\theta) \) or \( r = a \sin(n\theta) \). In our case, the equation given is \( r = 2 \cos 3\theta \). Here, 'a' is 2, which defines the length of each petal. The '3' specifies how many petals the rose will have since 'n' is odd. This means the curve will have three petals.

These curves are known as rose curves because they create patterns reminiscent of flower petals. When graphing these equations, you'll see petal-like shapes radiating symmetrically from the origin. Rose curves can vary in complexity depending on the numbers in the function, but this three-petaled version is relatively straightforward. Remember, when 'n' is even, the number of petals doubles. Exploring these curves can be a delightful way to understand symmetry and periodicity in polar graphs.

Polar Graph

Polar graphs are a way of displaying two-dimensional data using polar coordinates. Unlike the Cartesian coordinate system, which uses (x, y) for points, polar coordinates express locations using a radius \( r \) and angle \( \theta \).

In polar graphs, each point is defined by how far it is from the origin (radius) and the angle it makes with the positive x-axis. This creates a circular plotting space where curves often form intriguing spiral and petal shapes, just like in the exercise with the rose curve. Graphing in polar coordinates allows us to visualize equations like \( r = 2 \cos 3\theta \) in a way that highlights their unique cyclical and symmetrical nature.

• The center of the graph is called the pole (similar to the origin in Cartesian coordinates).
• Angles are typically measured in radians, though degrees can also be used.
• This type of plotting is excellent for capturing periodic or circular behaviors in data.

Understanding these basics can make the transition between Cartesian and polar plots much easier.

Symmetry in Polar Coordinates

Symmetry plays a crucial role in polar graphs, particularly when dealing with equations like the rose curve. In polar coordinates, symmetry can be observed in various forms:

• **Symmetry about the polar axis** - Often seen with functions involving cosine, such as our equation \(r = 2 \cos 3\theta\). This symmetry implies that the graph will look the same if you reflect it over the polar axis (like folding a circle along its diameter).
• **Symmetry about the line \(\theta = \frac{\pi}{2}\)** - Common in equations involving sine. This mirrors the curve across a vertical line through the pole.
• **Symmetry about the pole** - If the graph is unchanged when \( r \) is replaced with \(-r\), indicating that the curve reflects evenly about the origin.

For the equation \(r = 2 \cos 3\theta\), symmetry about the polar axis ensures each petal is evenly spaced forming this three-leaved rose curve. Understanding these symmetrical properties helps in efficiently sketching and analyzing polar graphs by predicting and verifying the curve's overall form.