Problem 47 - Calculus Volume 3

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. $$ \begin{array}{l} x=3 \cosh (4 t) \\ y=4 \sinh (4 t) \end{array} $$

Step-by-Step Solution

Answer

The parametric equations represent a hyperbola.

Step 1: Understanding the Parametric Equations

The given equations are in the form \(x = 3 \cosh(4t)\) and \(y = 4 \sinh(4t)\). Here, the functions \(\cosh\) and \(\sinh\) are hyperbolic functions, similar to trigonometric functions but for hyperbolas.

Step 2: Recognize the Forms of Hyperbolic Functions

Hyperbolic functions \(\cosh\) and \(\sinh\) are defined as \(\cosh(u) = \frac{e^u + e^{-u}}{2}\) and \(\sinh(u) = \frac{e^u - e^{-u}}{2}\). These functions are often used to define hyperbolas in parametric form.

Step 3: Identify the Equation of Hyperbola

The parametric equations for a hyperbola typically relate to the identity \(\cosh^2(u) - \sinh^2(u) = 1\). By examining the given equations, where \(x = a\cosh(kt)\) and \(y = b\sinh(kt)\), and referring to the identity, we recognize this relation forms a hyperbola.

Step 4: Recognize the Hyperbola

When the parametric forms involve \(a\cosh(kt)\) and \(b\sinh(kt)\), it signifies a hyperbola centered at the origin, oriented along the axes determined by constants \(a\) and \(b\). Here, \(a = 3\) and \(b = 4\), confirming this is a hyperbola.

Key Concepts

Hyperbolas

A hyperbola is a type of curve on a plane, similar to an ellipse but with distinct properties. Unlike ellipses, which are closed curves, hyperbolas consist of two separate branches. These branches mirror each other and open either left-right or up-down.

Hyperbolas are formed as part of the conic sections, a family that includes circles, ellipses, and parabolas. They occur when we slice a double cone with a plane in such a way that the angle of the plane is steeper than the side of the cone. This gives hyperbolas a characteristic look.

• If you imagine two mirror-image curves, that's a hyperbola.
• Their central focus points lead to exciting mathematical properties.
• They extend infinitely while staying roughly symmetrical.

Hyperbolic Functions

Hyperbolic functions, \[\cosh(u)\] and \(\sinh(u)\), are analogues of trigonometric functions but involve exponential expressions. They are basis of the names of these functions—think of them as the "hyperbolic cosines" and "hyperbolic sines." Here's a quick introduction to what these functions mean:

• \(\cosh(u)\): Defined as \(\frac{e^u + e^{-u}}{2}\), it operates similar to the cosine function.
• \(\sinh(u)\): Defined as \(\frac{e^u - e^{-u}}{2}\), it resembles the sine function.

In applications, especially in physics and engineering, hyperbolic functions describe a wide range of phenomena, such as the shape of a hanging cable in physics, known as a catenary.

Cosh and Sinh

The specific hyperbolic functions used in the parametric equations provided are \(\cosh(4t)\) and \(\sinh(4t)\). These functions are instrumental in forming the standard hyperbolic identity:
\[\cosh^2(u) - \sinh^2(u) = 1\]. This is akin to the trigonometric identity \(\cos^2\theta + \sin^2\theta = 1\), demonstrating their role in defining hyperbolas.

When you see expressions like \( x = 3 \cosh(4t) \) and \( y = 4 \sinh(4t) \), they are directly hinting towards a hyperbola centered at the origin:

• \(\cosh\) stretches horizontally, aligning to parameter 'a'.
• \(\sinh\) extends vertically, adhering to parameter 'b'.

These functions work together to form hyperbolic shapes when visualized, similar to trigonometric functions forming circular shapes.