Problem 43 - A Graphical Approach to Precalculus with Limits

Question

Graph each polar equation for \(\theta\) in \(\left[0^{\circ}, 360^{\circ}\right)\). In Exercises \(39-48\), identify the rype of polar graph. $$r=4 \cos 2 \theta$$

Step-by-Step Solution

Answer

The polar equation \( r = 4 \cos 2 \theta \) is a rose curve with 4 petals.

Step 1: Understanding the Equation

The given polar equation is \( r = 4 \cos(2\theta) \). This equation is in the form of \( r = a \cos(n\theta) \), which generally represents a rose curve. Here, \( a = 4 \) and \( n = 2 \).

Step 2: Determine the Type of Graph

Since \( n = 2 \), the equation \( r = 4 \cos(2\theta) \) will form a rose curve with \( 2n = 4 \) petals. The number of petals is even, which is determined by doubling \( n \).

Step 3: Calculate and Plot Sample Points

For a complete graph, calculate \( r \) for various values of \( \theta \) from \( 0^{\circ} \) to \( 360^{\circ} \): - \( \theta = 0^{\circ} \), \( r = 4 \cos(0) = 4 \)- \( \theta = 45^{\circ} \), \( r = 4 \cos(90^{\circ}) = 0 \)- \( \theta = 90^{\circ} \), \( r = 4 \cos(180^{\circ}) = -4 \)And continue this step until \( \theta = 360^{\circ} \) to verify symmetry and complete petal formation.

Step 4: Analyzing Graph Behavior and Symmetry

A rose curve of the form \( r = 4 \cos(2\theta) \) is symmetric about the x-axis. The calculation and plotting confirm the presence and structure of the four petals that are expected from the equation.

Key Concepts

Rose Curve

The rose curve is an enchanting pattern that emerges when graphing polar equations of the form \( r = a \cos(n\theta) \) or \( r = a \sin(n\theta) \). These curves are visually striking and quite popular in the study of polar coordinates.
In the given equation \( r = 4 \cos(2\theta) \), we clearly see the structure that represents a rose curve. This equation will form a rose-like shape when plotted on the polar graph, providing not just mathematical insight but also a beautiful visual pattern.
These rose curves appear like petals extending from the origin, creating an appealing flower-like structure that makes them fascinating to explore.

Petals

Understanding how petals form in a rose curve is crucial. In our example \( r = 4 \cos(2\theta) \), the number of petals is dictated by the parameter \( n \).
Let's break it down:

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

For our equation, since \( n = 2 \), it leads to a total of \( 4 \) petals because \( 2 \times 2 = 4 \). These petals are evenly spaced around the origin, creating a symmetrical appearance that is enchanting both mathematically and visually.

Symmetry

Symmetry is a significant attribute of rose curves, adding to their beauty and mathematical elegance. For our equation, \( r = 4 \cos(2\theta) \), the graph demonstrates symmetry about the x-axis.
This means:

• For each point on the curve, there is a corresponding point directly opposite it across the x-axis.
• This property simplifies graphing the equation, as half of the graph can be mirrored over the x-axis to obtain the full shape.

The symmetry property not only makes these curves aesthetically pleasing but also reduces the complexity involved in understanding and predicting their behavior.

Polar Graph

The polar graph is the canvas where these equations transform into beautiful patterns. Unlike the Cartesian plane, the polar coordinate system uses the radius \( r \) and angle \( \theta \) to define points.
In our case with \( r = 4 \cos(2\theta) \), points are plotted based on their distance from the pole (origin) and their angle from the positive x-axis. This results in the intriguing rose curve with 4 petals.
Analyzing such graphs helps in understanding the relationship between algebraic equations and their geometrical representations. It is essential in subjects like physics and engineering where circular and spiral paths are common.