Question

A family of curves called hyperbolas (discussed in Section 10.4 ) has the parametric equations \(x=a\) tan \(t\) \(y=b \sec t,\) for \(-\pi

Short answer

For the given parametric equations \(x = \tan t\) and \(y = \sec t\), we plotted points for selected values of \(t\) in the range \(-\pi < t < \pi\), avoiding \(t = \pm \pi/2\). We then connected the plotted points in the order they appear as \(t\) increases, showing the direction in which the hyperbola is generated. Finally, the connected points form the hyperbola curve with \(a = b = 1\).

Step-by-step solution

Write down the parametric equations

Given the parametric equations for hyperbola are \(x = a \tan t\) and \(y = b \sec t\), we need to plug in the values of \(a\) and \(b\) as 1. So we have: \(x = \tan t\) \(y = \sec t\)

Create a table of values containing \(t\), \(x\), and \(y\)

Let's generate some values of \(x\) and \(y\) for various values of \(t\) in the range \(-\pi < t < \pi\). Remember that \(|t| \neq \pi/2\), so we will exclude \(t = \pm \pi/2\) from the table. Choose, for example, \(t = -\pi + 0.1, -\pi/2 + 0.1, \pi/2 - 0.1, \pi - 0.1\): | \(t\) | \(x\) | \(y\) | |--------------|--------------|-------------| | \(-\pi + 0.1\) | \(\tan(\theta - \pi + 0.1)\) | \(\sec(\theta - \pi + 0.1)\) | | \(-\pi/2 + 0.1\) | \(\tan(\theta - \pi/2 + 0.1)\) | \(\sec(\theta - \pi/2 + 0.1)\) | | \(\pi/2 - 0.1\) | \(\tan(\theta + \pi/2 - 0.1)\) | \(\sec(\theta + \pi/2 - 0.1)\) | | \(\pi - 0.1\) | \(\tan(\theta + \pi - 0.1)\) | \(\sec(\theta + \pi - 0.1)\) | Calculate these values to obtain the points to plot.

Plot the points on the graph

Next, plot each point (\(x\), \(y\)) on a coordinate plane: 1. Point 1: \((\tan(\theta - \pi + 0.1), \sec(\theta - \pi + 0.1))\) 2. Point 2: \((\tan(\theta - \pi/2 + 0.1), \sec(\theta - \pi/2 + 0.1))\) 3. Point 3: \((\tan(\theta + \pi/2 - 0.1), \sec(\theta + \pi/2 - 0.1))\) 4. Point 4: \((\tan(\theta + \pi - 0.1), \sec(\theta + \pi - 0.1))\)

Indicate the direction of the curve

As \(t\) increases from \(t = -\pi\) to \(t = \pi\), draw arrows on the graph to visualize the direction in which the hyperbola is generated. You will see that the curve follows the order of the points: Point 1, Point 2, Point 3, and Point 4.

Connect the plotted points to form the curve

Now that we have plotted the points and indicated the direction in which the curve is generated, we can connect them to create the hyperbola curve. Note that the curve will not cross the asymptotes. The graph now shows the hyperbola with \(a = b = 1\).

Key concepts

Graphing Hyperbolas

Understanding hyperbolas begins with their distinctive shapes and the rules for graphing them. A hyperbola consists of two branches, which are mirror images, separated by a pair of lines called asymptotes.

When graphing hyperbolas with the equation given by the parametric equations \(x = a \tan t\) and \(y = b \sec t\), the first step is to identify the range for the parameter \(t\). Since the tangent and secant functions have undefined values (and hence the curve has asymptotes), we exclude \(t = \pm \frac{\pi}{2}\) from the range.

The next step involves plotting several points on the hyperbola by choosing values of \(t\) and calculating the corresponding \(x\) and \(y\) coordinates using the given equations. These points can then be plotted on the coordinate plane, and by connecting them, while respecting the asymptotes, the shape of the hyperbola emerges. It's also important to indicate the direction in which the curve is traced as the parameter \(t\) increases, often marked with arrows on the graph.

Trigonometric Identities

In the context of parametric equations, trigonometric identities are a set of equalities involving trigonometric functions that can simplify calculations and help in graphing curves. For instance, they can be used to transform the expressions of \(\tan t\) and \(\sec t\) or find relationships between different trigonometric functions based on the angle \(t\).

Some commonly used identities include \(\sec^2 t - \tan^2 t = 1\) and \(\tan(\frac{\pi}{2} \pm t) = \mp \cot t\). These identities are intrinsic to understanding the behavior of the hyperbola's parametric equations around the asymptotes. By applying these identities, one can avoid indeterminate forms that occur at the asymptotes and analyze the hyperbola's behavior at points close to \(t = \pm \frac{\pi}{2}\).

Parametric Equations

Parametric equations represent a set of equations where the system's variables are expressed as continuous functions of one or more independent parameters. In the case of a hyperbola, parametric equations allow us to express \(x\) and \(y\) coordinates in terms of a single parameter \(t\), which can be visually interpreted as time.

When graphing, we choose values for \(t\) and derive the corresponding \(x\) and \(y\) to generate points that outline the curve. Parametric equations are particularly useful for curves like hyperbolas, where the relationship between \(x\) and \(y\) is not a simple function and may include multiple values of \(y\) for a single \(x\) or vice versa. By using a parameter, we untangle this complexity and create a smooth representation of the curve.

Calculus

Calculus plays a central role when dealing with curves defined by parametric equations, especially when discussing properties such as tangents, areas, arc lengths, and curvature.

By using calculus, we can differentiate the parametric equations to find the slopes of the tangents at various points, thus understanding the steepness and direction of the curve. Additionally, we can integrate to find the area enclosed by a hyperbola or determine the length of a segment of the curve.

While the exercise provided doesn't delve into the full depths of calculus applications, understanding the fundamentals of differentiation and integration are vital for mastering the study of curves through parametric equations.