Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

The area of the lemniscate \(r^{2}=a^{2} \cos 2 \theta\) is \(a^{2} .\) Sketch the graph of \(r^{2}=16 \cos 2 \theta .\) Then find the area of one loop of the graph.

Step-by-Step Solution

Answer

The area of one loop of the graph \(r^{2}=16 \cos 2 \theta\) is 8.

Step 1: Sketching the Graph

A Lemniscate is a figure shaped like an infinity symbol, comes from the polar equation \(r^{2} = a^{2} \cos 2 \theta\). Here, \(a^{2}\) is given as 16. So, the function takes the form \(r^{2} = 16 \cos 2 \theta\). Plot its graph using polar coordinates.

Step 2: Recognizing the Polar Graph

After sketching, you will see that the graph has two symmetrical loops around the origin (0,0). Each loop's furthest point from the origin lies on the horizontal axis and at a distance of 'a' from the origin.

Step 3: Calculating Area of One Loop

Given that the area of the lemniscate is \(a^{2}\), and noting that Lemniscate has two identical loops, the area of one loop is \(\frac{a^{2}}{2}\). Here, \(a^{2}\) is 16. Therefore, the area of one loop is \(\frac{16}{2} = 8\).

Key Concepts

Polar Coordinates

Polar coordinates represent an alternative to Cartesian (or rectangular) coordinates for describing positions in the plane. In this system, a point's location is determined by a distance from a fixed point—usually called the pole or origin—and an angle measured from a fixed direction.

Here's a simple way to visualize polar coordinates: Imagine you're standing at the center of a circular track. The pole is where you stand, and the fixed direction could be directly in front of you, which we call the polar axis. To indicate a point on the track, you could say how far someone should walk from you (the radius, usually denoted as 'r') and the direction they should head (the angle, denoted by 'θ' or theta).

This polar coordinate system is especially useful for circles and shapes that feature symmetry about a point, making it perfect for analyzing figures like the lemniscate discussed in the lemniscate area calculation exercise.

Polar Equations

Polar equations are mathematical expressions that define curves in the plane using the polar coordinate system. One of the most remarkable characteristics of these equations is their ability to describe complex curves with relative simplicity.

For instance, the polar equation for a circle with radius 'a' centered at the origin is simply 'r = a'. This elegant expression could be contrasted with its Cartesian equation 'x^2 + y^2 = a^2', which is slightly more complex. Now, when it comes to the lemniscate—represented by the equation \(r^2 = a^2 \cos 2\theta\)—the beauty of polar equations truly shines. This equation indicates that the distance 'r' varies depending on the angle 'θ', which generates the intriguing infinity shape of the lemniscate.

The exercise provided involves such polar equations and highlights the intuitive way they can describe complex geometric shapes. With an understanding of polar equations, students can tackle the beauty of geometry in a new and sometimes more straightforward form.

Graph Sketching

Graph sketching is a vital skill for visualizing and understanding mathematical functions and relations. When it comes to polar equations, sketching the graph can often reveal the symmetry and intricacies of the curve in a way that the equation alone might not convey.

In the case of the lemniscate with equation \(r^2 = 16 \cos 2\theta\), sketching starts by plotting key points and noticing how 'r' changes with 'θ'. A powerful technique is to vary 'θ' from 0 to 2π radians and see where 'r' is positive, negative, or zero, which corresponds to the points where the curve intersects the pole. The two loops of the lemniscate become apparent as 'θ' sweeps across different quadrants, highlighting the figure's symmetry about the origin.

The step-by-step solution references graph sketching as the first step, indicating its importance in the overall understanding of the problem. Once the graph is sketched, determining properties like the area becomes more manageable since the visual representation aids in comprehending the spatial distribution and extent of the curve.