Problem 174 - Calculus Volume 3

Question

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

Step-by-Step Solution

Answer

The graph has symmetry about the polar axis, and it forms a curve with repeated patterns every \( 2\pi \).

Step 1: Understand the Equation

Identify the type of polar equation given. In this case, the equation is of the form \( r = a \cos(n\theta) \), which often represents a rose curve, except here \( n = \frac{1}{2} \) making it slightly atypical.

Step 2: Determine Symmetry

To check for symmetry, we apply the transformation rules: - Replace \( \theta \) with \( -\theta \). The equation becomes \( r = 3 \cos\left(-\frac{\theta}{2}\right) = 3 \cos\left(\frac{\theta}{2}\right) \). This confirms symmetry about the polar axis.- Check for pole symmetry by replacing \( r \) with \(-r\). This would convert to \(-r = 3 \cos\left(\frac{\theta}{2}\right)\), which is not satisfied by the original equation indicating no pole symmetry.- Symmetry about \( \theta = \frac{\pi}{2} \) involves replacing \( r \) with \(-r \) and \( \theta \) with \( \pi - \theta \). This also does not preserve the original form.

Step 3: Analyze the Range of \( \theta \)

Given the cosine function \( \cos\left(\frac{\theta}{2}\right) \), the output of cosine function ranges from \(-1\) to \(1\). Thus, \( r \) ranges from \(-3\) to \(3\), although negative \( r \)'s reflect the graph over the origin in polar coordinates.

Step 4: Determine the Key Angles and Sketch

To start sketching, compute several key points. For example, for \( \theta = 0 \), \( r = 3 \). For \( \theta = \pi \), \( r = 0 \), and periodic continuation shows repeat behavior for \( 2\pi \). Similarly, check points at \( \theta = \frac{\pi}{2}, \frac{3\pi}{2} \) where \( r \) would be \( \pm \sqrt{3} \). Plotting these points helps visualize the petals of the curve.

Step 5: Identify Symmetrical Divide

Given \( r = 3 \cos\left(\frac{\theta}{2}\right) \), the symmetry about the polar axis indicates the graph reflects perfectly on either side of the \( x \, -\) axis upon plotting.

Key Concepts

Symmetry in Polar Equations

Symmetry in polar equations is an important characteristic that helps simplify the graphing process by reducing the amount of computation needed. Symmetry can indicate that certain sections of a graph are mirror images of other sections, thus repeating patterns emerge.

• Polar Axis Symmetry: This type of symmetry occurs when a graph remains unchanged if you replace \( \theta \) with \( -\theta \). For the equation \( r = 3 \cos \left(\frac{\theta}{2}\right) \), replacing \( \theta \) with \( -\theta \) does not alter the equation's form, indicating symmetry about the polar or horizontal axis.


• Pole Symmetry: For an equation to show pole symmetry, switching \( r \) with \(-r\) must leave the graph unchanged. Here, this change leads to a different equation, showing there is no pole symmetry.


• \( \theta = \frac{\pi}{2} \) Symmetry: Typically checked by replacing \( \theta \) with \( \pi - \theta \) and \( r \) with \(-r\), but these steps do not preserve the form for our equation. Hence, this symmetry does not exist here.

Rose Curves

Rose curves are a special type of polar graph known for their petal-like appearance. They are expressed in the form \( r = a \cos(n\theta) \) or \( r = a \sin(n\theta) \), where \( n \) determines the number and arrangement of petals.

• For even \( n \), the rose curve has \( 2n \) petals.


• For odd \( n \), the rose curve has \( n \) petals.

However, the given equation \( r = 3 \cos \left(\frac{\theta}{2}\right) \) involves a unique twist where \( n = \frac{1}{2} \). This fractional value influences the shape subtly by affecting the symmetry and repetition.
The graph displays a two-petaled rose due to the period and nature of the cosine function involved. Analyzing the equation over a complete set of angles shows how the curve folds back on itself to form these beautiful patterns.

Graphing Polar Equations

Graphing polar equations like \( r = 3 \cos \left(\frac{\theta}{2}\right) \) involves determining key features that help plot the shape accurately. Unlike Cartesian coordinate plots, polar plots are generally created by evaluating the radius \( r \) at different angle measures \( \theta \).

• Identify Key Points: Start by choosing angles such as \( \theta = 0, \pi, \frac{\pi}{2}, \frac{3\pi}{2} \) to find the respective values of \( r \). Calculating these helps to locate important points on the graph.


• Consider Periodicity: For polar equations, periodic behavior will help transition the graph between repetitions smoothly. Here, observe how the curve repeats its pattern over \( 2\pi \) to form a rose-like shape.

Plotting these points and keeping in mind the symmetry observed, you connect them to sketch the graph. In this exercise, the graph reflects across the polar axis, displaying its symmetrical nature while forming a visually distinct rose curve.