Question

Sketch a graph of the polar equation and identify any symmetry. $$ r=3 \cos (2 \theta) $$

Short answer

The graph of the polar equation \( r = 3 \cos(2\theta) \) is a rose curve with 4 petals symmetric about the polar axis and the origin.

Step-by-step solution

Analyze Polar Equation

The polar equation given is \( r = 3 \cos(2\theta) \). This is a type of polar equation known as a rose curve. The parameter \( a \) is 3 and \( n \) is 2 in the form \( r = a \cos(n\theta) \). For \( \cos(n\theta) \) where \( n \) is even, the rose has \( 2n = 4 \) petals.

Determine Symmetry

Due to the cosine component, the equation is symmetric with respect to the initial line (polar axis). Cosine functions are even functions, meaning \( \cos(-\theta) = \cos(\theta) \), which implies symmetry about the line \( \theta = 0 \). Additionally, since \( n \) is even, the graph exhibits symmetry about the origin as well.

Calculate Key Points

To sketch the graph, identify key points by substituting \( \theta \) values. Substitute \( \theta = 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi \) and so on into \( r = 3 \cos(2\theta) \) to find zeroes and maximums of \( r \). For example, at \( \theta = 0 \), \( r = 3 \). At \( \theta = \frac{\pi}{4} \), \( r = 0 \), another petal starts to form.

Sketch the Graph

Using the key points and symmetry, sketch the graph. Start at \( \theta = 0 \) with \( r = 3 \), mark a point on the polar axis. As \( \theta \) increases, \( r \) becomes zero at \( \frac{\pi}{4} \). Continue marking points and join them. This creates a rose curve with 4 petals, 2 extending on the positive and negative x-axis, and 2 on the y-axis.

Key concepts

Understanding Rose Curves

A rose curve is a fascinating type of polar graph that has a unique, petal-like structure. These curves are defined by polar equations of the form \( r = a \cos(n\theta) \) or \( r = a \sin(n\theta) \). The number of petals in the curve depends on the value of \( n \).

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

In the case of \( r = 3 \cos(2\theta) \), since \( n \) equals 2, there are 4 petals.
Each petal represents a complete cycle of the function as \( \theta \) extends over its range, typically \( 0 \) to \( 2\pi \) for \( \cos \) functions. The maximum length of the petals is determined by \( a \), which in this scenario is 3, indicating that each petal extends out to a radius of 3.

Symmetry in Polar Coordinates

Symmetry plays a key role in simplifying the graphing of polar equations. For the rose curve given by \( r = 3 \cos(2\theta) \), symmetry helps reveal its shape without plotting every point. Several forms of symmetry are relevant in polar graphs:

• Line (Polar Axis) Symmetry: The graph is symmetric with respect to the polar axis because the cosine function is an even function. This means that \( \cos(-\theta) = \cos(\theta) \), ensuring that the graph looks identical on both sides of the polar axis.
• Origin Symmetry: The graph is also symmetric about the origin due to the even number of petals created by the even value of \( n \).

The symmetry properties ensure the rose curve not only forms balanced petals but also repeats its pattern predictably.

Steps for Graph Sketching

Graph sketching in polar coordinates is made easier by following a systematic approach. For \( r = 3 \cos(2\theta) \), recognize the symmetry to reduce the work of plotting the curve:
1. **Identify Key Points:** These are the angles where known actions take place — such as \( \theta = 0, \frac{\pi}{4}, \frac{\pi}{2},...\) and so forth. At these angles, substitute in \( r = 3 \cos(2\theta) \) to find zeros and peak values.
2. **Plot the Basic Points:** Begin at \( \theta = 0 \), where \( r = 3 \). Watch as \( \theta \) increases to \( \frac{\pi}{4} \) where \( r \) returns to zero; this is the start of a new petal.
3. **Leverage Symmetry:** Since the graph is symmetric about the polar axis and origin, you can mirror the plotted points in other sectors of the graph.
4. **Connect the Dots:** Once sufficient key points are marked, smoothly connect them forming the petals. Ensure the curved lines reflect the nature of the cosine function in polar form.

Trigonometric Functions in Polar Graphs

Trigonometric functions are fundamental in defining rose curves and other polar graphs. They determine the oscillation and repetition within a specific range of angles. In the curve \( r = 3 \cos(2\theta) \):

• The **Cosine Function:** Represents periodic repetition, completing a full cycle over \( 0 \) to \( 2\pi \). Being an even function, it naturally provides symmetry around the polar axis.
• The **Amplitude and Frequency:** Here, \( a = 3 \) defines how far each petal stretches from the center, while \( n = 2 \) determines how many times the pattern repeats (or how many petals are formed).

The interplay between \( a \) and \( n \) in the trigonometric equation is what produces the distinct petal structure of a rose curve, making it a popular choice for exploring polar graphs in mathematics.