Question

Sketch a graph of the polar equation and identify any symmetry. $$ r^{2}=4 \cos (2 \theta) $$

Short answer

The graph is a lemniscate, symmetric about the polar axis and the origin.

Step-by-step solution

Understanding the Equation

The given polar equation is \( r^2 = 4 \cos(2\theta) \). This is a type of polar equation known as a "lemniscate." The presence of \( \cos(2\theta) \) suggests potential symmetry and the specific form indicates a lemniscate of Bernoulli.

Convert to Cartesian Coordinates

To better visualize the equation, convert it to Cartesian coordinates: Start by using the formulas \( x = r \cos \theta \) and \( y = r \sin \theta \). The identity \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \) can help transition to Cartesian form. Substitute \( r^2 = x^2 + y^2 \) and simplify.

Test for Symmetry

To test for symmetry in polar coordinates, consider three types: polar-axis (\( \theta \to -\theta \)), line \(\theta = \frac{\pi}{2}\) (\( r \to -r \)), and pole symmetry (\( \theta \to \pi - \theta \)). Check each by substituting into the equation and confirming if the original form is restored.

Identifying the Symmetry

Substituting \( \theta \to -\theta \) in \( r^2 = 4 \cos(2\theta) \), we get \( r^2 = 4 \cos(-2\theta) \), which is the same as the original equation, confirming symmetry about the polar axis. Similarly, the symmetry is unchanged when \( r \to -r \), indicating symmetry about the origin.

Sketch the Graph

Plot the lemniscate directly using the polar equation. Consider key angles like \( \theta = 0, \frac{\pi}{4}, \frac{\pi}{2}, \pi \) to determine the behavior of the graph in different directions. The lemniscate should resemble an infinity symbol \( \infty \) centered at the pole.

Viewing the Complete Graph

After sketching the basic loops for several angles, complete the graph by demonstrating full oval-like loops that represent the lemniscate structure characteristic of this form. The full graph should show symmetry about the polar axis and the origin.

Key concepts

Symmetry

Symmetry in polar equations provides valuable insights into the behavior of the graph and can simplify the process of sketching it. For the given equation \( r^2 = 4 \cos(2\theta) \), testing for symmetry involves substituting certain transformations of \( \theta \) and \( r \).

Three primary types of symmetry in polar coordinates are:

• Polar-Axis Symmetry: This is tested by substituting \( \theta \to -\theta \). If the equation remains unchanged, the graph is symmetric about the polar axis (the horizontal axis passing through the pole).
• Line Symmetry (\(\theta = \frac{\pi}{2}\)): This involves checking what happens when you substitute \( r \to -r \). If the equation holds the same structure, it indicates symmetry about the vertical line at \(\theta = \frac{\pi}{2}\).
• Pole Symmetry: Substituting \( \theta \to \pi - \theta \) can confirm this type of symmetry. If the equation remains unchanged, the graph has symmetry about the pole itself.

For the lemniscate equation, checking these symmetries confirms polar-axis symmetry and symmetry about the origin, making it easier to visualize and sketch the graph.

Lemniscate of Bernoulli

The lemniscate of Bernoulli is a well-known type of curve in polar coordinates resembling the infinity symbol \( \infty \). It is defined by equations similar to the form \( r^2 = 4 \cos(2\theta) \).

Here are some important characteristics of a lemniscate of Bernoulli:

• A lemniscate is centered at the pole (the origin in polar coordinates) and extends outward in loops.
• The equation typically involves a trigonometric function like \( \cos(2\theta) \) or \( \sin(2\theta) \), hinting at the doubled polar angle which affects the shape.
• The constant on the right side of the equation affects how stretched the loops are, with larger constants resulting in the larger resulting loop span.

Understanding these characteristics helps in identifying the structure of the graph and anticipating where each loop will be positioned relative to the pole.

Polar Graph Sketching

Sketching polar graphs can sometimes appear challenging, but by understanding the underlying equations and their symmetry, it becomes easier. For graphing the lemniscate \( r^2 = 4 \cos(2\theta) \), use some strategic angles to plot:

• Key angles: Begin with \( \theta = 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi \) as starting points to determine the general direction and shape of the graph.
• Plotting points: For each of these angles, calculate the corresponding \( r \) value and mark the point on the polar plane.
• Connecting the dots: Plot points might sketch a loop, characteristic of a lemniscate. As you sketch these loops, ensure the accuracy of symmetry across the polar-axis and origin.

The vital part is to complete an accurate graph where the loops intersect the pole and form the infinity-like shape around it. Once points are in place, the curve is the essence of exploring polar graph symmetry and aesthetics in math.