Problem 69 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 45-78: Sketch the graph of the polar equation. $$ r^{2}=4 \cos 2 \theta \quad \text { (lemniscate) } $$

Step-by-Step Solution

Answer

Sketch a symmetric figure-eight shape about the pole using the equation \( r^2 = 4\cos 2\theta \).

Step 1: Understand the Polar Equation

The given polar equation is \( r^2 = 4 \cos 2\theta \). This is the equation of a lemniscate, which typically forms a figure-eight shape in the polar coordinate system.

Step 2: Analyze the Equation Symmetry

The equation \( r^2 = 4 \cos 2\theta \) indicates that the graph will be symmetric with respect to the polar axis (the line \( \theta = 0 \)), since \( \cos 2\theta \) is an even function.

Step 3: Identify Points when \( r = 0 \)

Setting \( r = 0 \), we have \( 0 = 4 \cos 2\theta \), which implies \( \cos 2\theta = 0 \). Thus, \( 2\theta = \frac{\pi}{2} + n\pi \) (for odd integers \( n \)), leading to points where the graph passes through the pole, such as \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{3\pi}{4} \).

Step 4: Sketch Points using Key Angles

Calculate other key points by choosing angles within the range \( 0 \leq \theta < 2\pi \). For \( \theta = 0 \), \( r^2 = 4 \) gives \( r = \pm2 \). Similarly, find that for \( \theta = \frac{\pi}{2}, r = 0 \), forming the basis for the lobes of the lemniscate.

Step 5: Plot and Connect Points

Draw the symmetry of the lemniscate by connecting calculated points, ensuring that the graph forms a figure-eight on the polar axis centered at the pole. Two loops, one in the positive and one in the negative radial direction, will produce the iconic shape of the lemniscate.

Key Concepts

Lemniscate

A lemniscate can be best understood as a figure-eight shape in the polar coordinate system. It is like a sideways infinity symbol. The equation that forms a lemniscate is typically in the form of \(r^2 = a^2 \cos(2\theta)\) or \(r^2 = a^2 \sin(2\theta)\). This curve is unique because it is symmetrical and passes through the pole, making it visually distinct and mathematically interesting.When you sketch this shape in polar coordinates, it appears as two loops that touch at the origin, the point where the radius, \(r\), is zero. This contact point is crucial as it divides the lemniscate into its two loops. The lemniscate is often a topic of curiosity because of its aesthetic shape and symmetric properties which we will explore further.

Polar Equation

Polar equations represent curves in a coordinate system using the distance of the point from the origin (radius \(r\)) and the angle \(\theta\) from a reference direction, typically the positive x-axis in Cartesian coordinates. The given equation \(r^2 = 4 \cos 2\theta\) defines the polar coordinates, where

• \(r\) is the radial distance from the origin.
• \(\theta\) is the angular position.

In our specific equation of the lemniscate, the trigonometric function \(\cos 2\theta\) governs how \(r\) changes with \(\theta\), reflecting the periodic and symmetric nature of the shape.Because this equation uses a trigonometric function, it is important to recognize the evenness of \(\cos\) that results in the symmetry of the lemniscate. The presence of \(2\theta\) means the cycle repeats twice as \(\theta\) goes from 0 to \(2\pi\), affecting the pattern of the graph.

Graph Symmetry

Graph symmetry is a fascinating feature in polar coordinates, particularly with lemniscates. The symmetry can be evaluated in multiple ways:

• Polar Axis Symmetry: Here, the equation shows that \(r^2 = 4 \cos 2\theta\) remains unchanged if \(\theta\) is replaced by \(-\theta\), further indicating symmetry about the polar axis (the initial line or x-axis).
• Origin Symmetry: The lemniscate also displays central symmetry, being identical when viewed from opposite directions from the origin. This reflection property contributes to its figure-eight shape.

To verify symmetry, substituting opposite angle values into the equation can show whether the equation remains valid, reinforcing the curve's balanced nature in the plane around its pole.

Polar Axis

The polar axis acts much like the reference point for polar equations, similar to the x-axis in Cartesian coordinates. It is the line where \(\theta = 0\). When observing polar curves, the polar axis is crucial for identifying symmetry and understanding graph characteristics.In the equation \(r^2 = 4 \cos 2\theta\),

• The fact that the cosine function (\(\cos\)) is even provides insight into how the graph is mirrored about this axis.
• It ensures the lemniscate forms symmetrically, folding over the polar axis.

Understanding how this works can make plotting polar curves easier as it provides you with a reference line to visualize transformations and predict the mirror image of plotted points.The polar axis not only serves as a line of symmetry but also as a guide to verify the accuracy of plotted points and the overall curvature of the graphs.