Problem 55 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the curve represented by the parametric equations. Witch of Agnesi: \(x=2 \cot \theta, y=2 \sin ^{2} \theta\)

Step-by-Step Solution

Answer

The graph of the Witch of Agnesi curve based on the given parametric equations will start vertical asymptotic line at \(x=2\) for \(\theta = -\(\pi /2\)\), peak at \((2,2)\) when \(\theta = 0\), and return to the vertical asymptotic line \(x=2\) as \(\theta\) approaches \(\pi /2\).

Step 1: Understand the parametric equations

The given parametric equations are \(x=2 \cot \theta\) and \(y=2 \sin ^{2} \theta\). While \(\cot \theta\) is the reciprocal of the tangent function (which gives the x-coordinate), \(\sin ^{2} \theta\) squares the output of the sine function (determines the y-coordinate).

Step 2: Compute values

For various \(\theta\) values ranging from -\(\pi /2\) to \(\pi /2 \), compute the \(x\) and \(y\) values using the given parametric equations. For instance, when \(\theta = 0\), \(x=2 \cot 0 = Infinity\) and \(y=2 \sin ^{2} 0 = 0\). At \(\theta = \pi /4\), \(x=2 \cot (\pi /4) = 2\) and \(y=2 \sin ^{2}(\pi /4) = 1\). Similarly, get values for more points.

Step 3: Plot the graph

Using a graphing utility, plot the points obtained from the previous step. The points will start from the vertical asymptotic line \(x=2\) at \(\theta = -\(\pi /2\)\), peak at \((2,2)\) when \(\theta = 0\), and trace back along the curve to the vertical asymptotic line \(x=2\) as \(\theta\) approaches \(\pi /2\). This will form the Witch of Agnesi curve.