Question

Eliminate the parameter in the given parametric equations. \(x=\cosh t, \quad y=\sinh t\)

Short answer

The parameter is eliminated to produce the hyperbola equation: \(x^2 - y^2 = 1\).

Step-by-step solution

Understanding Hyperbolic Functions

The given parametric equations are based on hyperbolic functions: \(x = \cosh t\) and \(y = \sinh t\). These relate to the identity \( \cosh^2 t - \sinh^2 t = 1 \). We will use this identity to eliminate the parameter \(t\).

Square Both Equations

First, square both parameter equations to derive expressions for \(\cosh^2 t\) and \(\sinh^2 t\). This results in \(x^2 = \cosh^2 t\) and \(y^2 = \sinh^2 t\).

Apply Hyperbolic Identity

According to the hyperbolic identity, we have \(\cosh^2 t - \sinh^2 t = 1\). Substitute the squared expressions from step 2 into this identity to form the equation: \(x^2 - y^2 = 1\).

Rewrite as Cartesian Equation

The resulting equation \(x^2 - y^2 = 1\) is the Cartesian equation of the hyperbola related to the given parametric equations, free of the parameter \(t\).

Key concepts

Parametric Equations

Parametric equations are a fascinating way of expressing mathematical equations, offering a unique perspective on graphs and shapes. They involve one or more variables, known as parameters, to describe a set of values. In most cases, the parameter, often denoted as \(t\), represents time or some other variable, allowing us to track how each value changes over a certain range. For example, the parametric equations \(x = \cosh t\) and \(y = \sinh t\) describe a curve on the coordinate plane by representing each point \((x, y)\) as a function of \(t\).
These equations are essential when analyzing curves that cannot be easily expressed with a single equation in the xy-plane. By breaking down complex curves into simpler components, parametric equations make it possible to represent intricate shapes and motions, ranging from circles to hyperbolas and beyond.

Hyperbolic Identity

The hyperbolic identity is a crucial element when dealing with hyperbolic functions like \(\cosh\) and \(\sinh\). The fundamental hyperbolic identity is \( \cosh^2 t - \sinh^2 t = 1 \), closely resembling the well-known Pythagorean identity, \(\cos^2 t + \sin^2 t = 1\), but with a different sign between the terms. This identity serves as a powerful tool in simplifying and solving equations related to hyperbolic functions.
In the parametric equations \(x = \cosh t\) and \(y = \sinh t\), this identity helps in a significant way. By squaring both sides of these equations, we derive \(x^2 = \cosh^2 t\) and \(y^2 = \sinh^2 t\). When substituted into the hyperbolic identity, it leads directly to a simplified version, which eliminates the parameter \(t\).
Understanding and using the hyperbolic identity are essential steps in transitioning from parametric to Cartesian equations, especially in the context of curves such as hyperbolas.

Cartesian Equation

A Cartesian equation is a conventional way of representing a curve by relating x and y directly on the coordinate plane without involving a parameter. It gives us a clear picture of the relationship between the two variables. In our exercise, after eliminating the parameter \(t\), the resulting Cartesian equation is \(x^2 - y^2 = 1\). This equation describes a hyperbola, a distinct type of curve with characteristic properties.
This particular hyperbola opens horizontally because the \(x^2\) term is positive while the \(y^2\) term is negative. Such equations are simpler to analyze graphically because they follow the familiar Cartesian coordinate system. Recognizing and converting parametric equations to their Cartesian counterparts allow for easier visualization and understanding of the geometric properties from the expressions.

Elimination of Parameter

Elimination of the parameter is a critical process when working with parametric equations. The goal is to remove the parameter, \(t\), from the equations to find a relationship that involves only \(x\) and \(y\). This simplifies the expression and allows us to see the curve's shape more directly.
In the given problem, we start by squaring the expressions for \(x\) and \(y\), arriving at \(x^2 = \cosh^2 t\) and \(y^2 = \sinh^2 t\). Subsequently, substituting these into the hyperbolic identity \(\cosh^2 t - \sinh^2 t = 1\), we succeed in eliminating \(t\) and derive the Cartesian equation \(x^2 - y^2 = 1\).
This step is vital as it simplifies parametric descriptions and helps in analyzing the geometric nature of curves. Whether in algebraic form or visually represented on the graph, eliminating the parameter is a fundamental skill in understanding and manipulating parametric equations.