Problem 23 - Precalculus with Calculus Previews

Question

Graphically show the difference between the given curves. \(x^{2}-y^{2}=1\) and \(x=\cosh t, y=\sinh t\)

Step-by-Step Solution

Answer

Both are the same hyperbola; graphically, they overlap exactly.

Step 1: Understanding the First Curve

The first equation given is \(x^2 - y^2 = 1\). This is the equation of a hyperbola, which is a type of conic section. Specifically, it represents a hyperbola centered at the origin (0,0) with its transverse axis along the x-axis.

Step 2: Understanding the Parametric Form

The second set of expressions are \(x = \cosh t\) and \(y = \sinh t\). These parametric equations describe a curve. Here, \(\cosh t\) and \(\sinh t\) are hyperbolic functions called hyperbolic cosine and hyperbolic sine, respectively.

Step 3: Relating Parametric Equations to the Hyperbola

For hyperbolic functions, there's an identity: \(\cosh^2 t - \sinh^2 t = 1\). Substituting \(x = \cosh t\) and \(y = \sinh t\) into the hyperbola equation \(x^2 - y^2 = 1\), we can verify that the parametric equations satisfy the hyperbola equation. Hence, \(x = \cosh t\) and \(y = \sinh t\) is a parametrization of the same hyperbola given by \(x^2 - y^2 = 1\).

Step 4: Graphical Representation

To graphically show the difference, plot the hyperbola given by \(x^2 - y^2 = 1\) using Cartesian coordinates, drawing two symmetrical curves opening right and left. The parametric form of \(x = \cosh t\), \(y = \sinh t\) will trace the same hyperbola as it defines the same curve.

Step 5: Conclusion of Graphical Differences

Both are representations of a hyperbola, which means there's no difference in shape or location between the given curve as an equation and as a parametrization. In a graph, they overlap completely, as the parametric form traces the same shape represented by the hyperbola equation.

Key Concepts

Hyperbola

A hyperbola is a type of curve that falls under the category of conic sections. It is defined by the equation \(x^2 - y^2 = 1\), which is known as a hyperbolic equation. A hyperbola consists of two separate curves called branches, which mirror each other through the center of the hyperbola.
The branches open out indefinitely and are shaped like a pair of opposing 'C's. In the case of the hyperbola characterized by \(x^2 - y^2 = 1\), the branches extend along the x-axis.The center of the hyperbola is located at the origin \(0,0\) in Cartesian coordinates. The two axes of a hyperbola are known as the transverse axis, which is the axis along which the hyperbola opens, and the conjugate axis, which is perpendicular to it. For our hyperbola, the transverse axis is the x-axis.

• The hyperbola is symmetrical along the transverse axis.
• Each branch approaches two asymptotes as it extends further which signifies the direction of its opening.

Conic Sections

Conic sections are shapes or figures that are formed by the intersection of a plane with a right circular cone. These include circles, ellipses, parabolas, and hyperbolas. Each has unique mathematical properties and equations.In the realm of conic sections, a hyperbola is recognized for its two distinct "U" shaped branches that are oriented in opposite directions. The hyperbola features prominently in fields such as physics and engineering due to its reflective properties.

• Circles and ellipses have no branches and are closed curves.
• Parabolas open infinitely wide in one direction.
• Hyperbolas, like the equation \(x^2 - y^2 = 1\), open in two directions, creating a distinct shape.

Understanding hyperbolas as conic sections helps differentiate them from the other types of conics by their open-ended nature and the way they graphically appear on a plane.

Hyperbolic Functions

Hyperbolic functions are analogs of the ordinary trigonometric functions but for a hyperbola, rather than a circle. Notably, they include hyperbolic sine \(\sinh t\) and hyperbolic cosine \(\cosh t\), which are crucial for describing hyperbolas parametrically.
These functions can be particularly useful in providing alternative ways to define and work with hyperbolas.The hyperbolic functions have the identity formula:\[\cosh^2 t - \sinh^2 t = 1\]This identity is starkly similar to the well-known Pythagorean identity for trigonometric functions \(\cos^2 \theta + \sin^2 \theta = 1\). Using this, we can parametrize hyperbolas effortlessly.

• \(\cosh t\) corresponds to the x-component of the hyperbola, showing the horizontal traversal.
• \(\sinh t\) corresponds to the y-component, showing vertical traversal.

By applying the parametric equations \(x = \cosh t\) and \(y = \sinh t\), we can derive the hyperbola \(x^2 - y^2 = 1\) quickly. These parametric expressions simplify the process of plotting hyperbolas on a graph, offering a dynamic way to represent them as compared to purely algebraic forms.