Problem 80 - Algebra and Trigonometry

Question

Use a graphing utility to obtain the plane curve represented by the given parametric equations. Witch of Agnesi: \(x=2\) cot \(t, y=2 \sin ^{2} t\) \([-6,6,1] \times[-4,4,1], 0 \leq t<2 \pi\)

Step-by-Step Solution

Answer

The plot obtained from the graphing utility should resemble an 8-shaped curve, which is known as the Witch of Agnesi curve. This is achieved by translating the parametric equations into polar form and then using the range and scales provided to graph the curve.

Step 1: Understanding Parametric Equations

We have a set of parametric equations, which define a plane curve in terms of the parameter t. Here, the equations are \(x=2\) cot \(t\) and \(y=2 \sin ^{2} t\). This means each point on our curve will have an x-coordinate and y-coordinate calculated from these equations, using the same value of t.

Step 2: Converting to Polar Coordinates

Typically, plane curves are described in a two-dimensional coordinate system, either in Cartesian coordinates (x,y) or polar coordinates (r,θ). Here, the given equations can be transformed into polar coordinates, where the angle is given by t, and radius by r. We can derive the transformation using the identities \(r = \sqrt{x^2+y^2}\) and \(\theta = \arctan(y/x)\). Using the equations given, we establish that \(r = \sqrt{(2 \cdot \cot(t))^2 + (2 \cdot \sin^2(t))^2}\), and \(\theta = t\).

Step 3: Graphing the Curve

Plot our radial function \(r(t)\) in polar coordinates using a graphing utility. Set the “x min” and “y min” to -6 and -4, respectively. Set “x max” and “y max” to 6 and 4, respectively. Let “x scl” and “y scl” be 1. Lastly, let t range from \(0 \leq t < 2 \pi\). This will give us a visual representation of the Witch of Agnesi in the two-dimensional plane.

Key Concepts

Plane Curve Graphing

Graphing plane curves is a fundamental aspect of understanding complex mathematical functions and their geometrical representations. A plane curve is defined as a curve that lies on a single plane, and it can take various forms, from straight lines to elaborate shapes like ellipses, parabolas, and more intricate figures such as the Witch of Agnesi.

In graphing plane curves, parametric equations play a pivotal role by defining each point on the curve with a set of equations dependent on one parameter. Typically, the parameter is denoted by 't' and varies over an interval, which results in different points (x, y) on the curve. For example, with the parametric equations provided, each x and y value is calculated by plugging in the values of t, giving us a collection of points that make up the curve when plotted.

The benefits of using parametric equations include the ability to represent motion and direction on a curve and to describe complex curves that may be challenging to express in standard x-y form. When graphing these curves by hand or with a utility, one creates points by calculating the x and y values for different t's and then plotting these points to visualize the shape of the curve. By connecting these points smoothly, one outlines the complete figure the parametric equations represent.

Polar Coordinates

Polar coordinates offer a unique and powerful way to represent points on a plane, particularly well-suited for graphs of circular or radial symmetry. The polar coordinate system specifies each point by a distance from a central point (known as the radius, r) and an angle (θ, theta) from a fixed direction, often the positive x-axis.

To understand and convert parametric or Cartesian coordinates to polar coordinates, we use the relationships: the radial distance r is found using the equation \( r = \sqrt{x^2 + y^2} \), and the angle θ by the arctangent of y over x, given by \( \theta = \arctan(y/x) \). These transformations allow us to reinterpret the curve's shape in terms of its radial distance from the origin and its angular direction, which can simplify the process of graphing certain curves.

Using polar coordinates, we can easily graph equations like the Witch of Agnesi, transforming them into a format that reveals their intrinsic symmetry and patterns that might not be as obvious in Cartesian coordinates. When graphing, one varies the angle θ, and the radius r is determined by the corresponding polar equation, plotting a curve that spirals out from the origin accordingly.

Witch of Agnesi

The Witch of Agnesi is an intriguing curve with a rich history in mathematics, named after Maria Gaetana Agnesi, an 18th-century Italian mathematician. It represents a quintessential example of a curve described using parametric equations.

Frequently used to illustrate the application of calculus and analytical geometry, the Witch of Agnesi can be described parametrically as given in our exercise: \(x = 2\) cot \(t\), \(y = 2 \sin^2(t)\). These equations generate an elegant, bell-shaped curve, symmetric about the y-axis and with a characteristic peak that flattens out as it moves away from the origin.

To graph the Witch of Agnesi effectively, you should understand the behavior of the trigonometric functions involved in its equations. The sine squared term in the y equation ensures that y is always non-negative and the cotangent determines the x values as the parameter t varies from 0 to \(2\pi\). By systematically calculating and plotting these values using the provided steps, one constructs the smooth, continuous curve that epitomizes the Witch of Agnesi, showcasing the power of parametric equations in describing complex shapes.