Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(1-32\), sketch the graph of the given polar equation and verify its symmetry (see Examples 1-3). \(r^{2}=4 \cos 2 \theta\) (lemniscate)

Step-by-Step Solution

Answer

The graph is a symmetric lemniscate about all axes.

Step 1: Understanding the Equation

We begin by analyzing the given polar equation \(r^2 = 4 \cos 2\theta\). This is a polar equation of a lemniscate, a figure-eight shape. We need to convert this into a basic understanding of the function before proceeding with graphing.

Step 2: Transforming for Graphing

Rearrange the equation for easier graphing in polar coordinates: \(r^2 = 4 \cos 2\theta\). Solve for \(r\) to get \(r = \pm \sqrt{4 \cos 2\theta}\). This gives us the radii for different angles \(\theta\).

Step 3: Plotting Key Points

Identify key angles, such as where \(\theta = 0, \pi/4, \pi/2,...\) and so on. For example, at \(\theta = 0\), \(r = \pm2\); at \(\theta = \pi/4\), \(r = 0\). Plot these points, noting that the graph will extend symmetrically in all four quadrants.

Step 4: Checking Symmetry

Verify symmetry about the polar axis, line \(\theta = \pi/2\), and the origin. For polar axis symmetry, replace \(\theta\) with \(-\theta\), yielding the same equation. Similarly, the substitution \(\theta\) with \(\pi - \theta\) for line symmetry and \(r\) with \(-r\) for origin symmetry confirm this symmetry.

Step 5: Drawing the Graph

Using the points and the symmetry properties, draw the lemniscate. It will show two loops at \(\theta = 0\) and \(\theta = \pi\), making a figure-eight shape.

Key Concepts

Lemniscate

The lemniscate is a fascinating geometric figure resembling a figure-eight. In polar coordinates, it is commonly expressed through equations involving trigonometric functions, such as \( r^2 = 4 \cos 2\theta \). This particular form of equation defines the shape and dimensions of the lemniscate.

To understand a lemniscate, imagine drawing a curve that loops symmetrically around the origin. These loops, seen clearly in the polar coordinate system, intersect at the origin point. This crossing creates the distinctive figure-eight appearance.

When graphing an equation like \( r^2 = 4 \cos 2\theta \), you'll notice the role of trigonometric functions. Here, the function \( \cos \) dictates how the loops form. Notably, lemniscates illustrate the beauty of polar coordinates, showing how complex and beautiful patterns arise from algebraic expressions.

Symmetry in Mathematics

Symmetry is a crucial concept in mathematics, offering insights into aesthetic patterns and simplifying mathematical analysis. When plotting polar equations, symmetry helps predict and verify graph shapes without computing every point. Instead, focus on different types of symmetry: polar axis, line, and origin symmetry.

**Polar Axis Symmetry:** This symmetry occurs if an equation remains unchanged when you replace \( \theta \) with \( -\theta \). For instance, in the equation \( r^2 = 4 \cos 2\theta \), swapping these values doesn’t alter the equation, showing polar axis symmetry.

**Line Symmetry:** If the form is unchanged when \( \theta \) becomes \( \pi - \theta \), the graph is symmetrical about a line, typically the vertical line at \( \theta = \pi/2 \).

**Origin Symmetry:** Occurs when replacing \( r \) with \( -r \) yields an identical equation. Verify your lemniscate here by recognizing that it retains its shape even when "flipped" over the origin.

These symmetrical properties facilitate quick and accurate graphing of curves and patterns, making them indispensable tools in mathematical graph analysis.

Graphing Polar Equations

Graphing in polar coordinates provides a unique insight into mathematics, focusing on angles and distances from a central point, rather than x and y coordinates. Here, the radii are key, transforming trigonometric expressions into curves and shapes.

Take the polar equation \( r^2 = 4 \cos 2\theta \). To graph it:

• **Solve for \( r \):** Rearrange to \( r = \pm \sqrt{4 \cos 2\theta} \). This shows how radius varies with angle \( \theta \).
• **Identify Key Angles:** Choose key values for \( \theta \), like \( 0 \), \( \pi/4 \), \( \pi/2 \), etc., to understand how \( r \) behaves.
• **Plot Points Symmetrically:** Use symmetry to extend your plot across quadrants, drawing a smooth continuous curve.
• **Draw the Graph:** Visualize the complete lemniscate, ensuring symmetry and accurate loop formation based on previously calculated points.

Remember, graphing polar equations like lemniscates offers an engaging challenge, blending algebra, trigonometry, and geometry into visually compelling forms.