Question

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{aligned} &x=\cosh t \\ &y=\sinh t \end{aligned} $$

Short answer

The equations represent a hyperbola.

Step-by-step solution

Identify the Variables in Parametric Equations

The given parametric equations are \(x = \cosh t\) and \(y = \sinh t\). These equations involve hyperbolic functions \(\cosh t\) and \(\sinh t\), which are similar to the trigonometric functions but for hyperbolas.

Recall the Relationship Between Cosh and Sinh

In hyperbolic functions, there is an identity similar to the Pythagorean identity for sine and cosine. This identity is \(\cosh^2 t - \sinh^2 t = 1\). This is crucial for identifying the type of curve represented.

Substitute Parametric Equations into Identity

Substitute \(x = \cosh t\) and \(y = \sinh t\) into the identity: \(\cosh^2 t - \sinh^2 t = 1\). Rewriting gives us \(x^2 - y^2 = 1\).

Determine the Type of Curve

The equation \(x^2 - y^2 = 1\) is the standard form of a hyperbola. Hence, the parametric equations \(x = \cosh t\) and \(y = \sinh t\) represent a hyperbola.

Key concepts

Hyperbolic Functions

Hyperbolic functions arise naturally in the context of describing curves known as hyperbolas, much like how trigonometric functions describe circles. Two primary hyperbolic functions are the hyperbolic cosine, denoted as \( \cosh t \), and the hyperbolic sine, denoted as \( \sinh t \). These functions are defined as follows:

• \( \cosh t = \frac{e^t + e^{-t}}{2} \)
• \( \sinh t = \frac{e^t - e^{-t}}{2} \)

These definitions are reminiscent of the exponential function and provide valuable insights into the geometry involved, especially through curves described by parametric equations using them.The importance of hyperbolic functions is not just theoretical; these functions model hyperbolic structures in the physical world, such as the shape of a hanging cable, known as a catenary. A key identity involving hyperbolic functions is \( \cosh^2 t - \sinh^2 t = 1 \), analogous to the trigonometric identity \( \cos^2 t + \sin^2 t = 1 \). This identity helps express hyperbolic relationships in terms of algebraic equations.

Identity in Parametric Equations

Identifying the relationship between parametric equations involves understanding how different functions work together to form specific curves. When dealing with equations such as \( x = \cosh t \) and \( y = \sinh t \), it's helpful to remember the identity \( \cosh^2 t - \sinh^2 t = 1 \). This identity is crucial in establishing that the parametric curve is a hyperbola.
By substituting the given parametric equations into this identity, you'll find that:

• From \( x = \cosh t \), we have \( x^2 = \cosh^2 t \).
• From \( y = \sinh t \), we have \( y^2 = \sinh^2 t \).

Consequently, the identity transforms into \( x^2 - y^2 = 1 \). This equation matches the standard form of a hyperbola, confirming the curve's identity. Recognizing such identities enables the translation of parametric descriptions into comprehensible algebraic equations, facilitating the interpretation of complex curves.

Types of Curves in Mathematics

In mathematics, various curves can be described using parametric equations. These include lines, circles, parabolas, ellipses, and hyperbolas. Each type projects a unique shape and has distinct algebraic properties.

• Lines: Represented typically by linear functions in their parametric form, combining consistent slopes without curvature.
• Circles and Ellipses: The circle is represented by trigonometric parametric equations, like \( x = \cos t \) and \( y = \sin t \), defining a closed loop. Ellipses extend circles into elongated shapes, often shown with equations like \( x = a \cos t \), \( y = b \sin t \).
• Parabolas: Often an element is squared in these equations, such as \( x = t^2 \) and \( y = t \), showing symmetry and a vertex from where they expand.
• Hyperbolas: Characterized by equations like \( x = \cosh t \), \( y = \sinh t \), their openness contrasts with other closed curves, marking them as unique due to their disconnected arms.

Recognizing these forms allows one to convert parametric equations to their spatial representation, facilitating better understanding of their geometric nature and significance in various applications.