Problem 432 - Calculus Volume 1

Question

For the following exercises, solve each problem. Prove that \((\cosh (x)+\sinh (x))^{n}=\cosh (n x)+\sinh (n x)\).

Step-by-Step Solution

Answer

\((\cosh(x)+\sinh(x))^n = \cosh(nx)+\sinh(nx)\) is proven using exponential identities.

Step 1: Understand Hyperbolic Functions

The hyperbolic functions are defined as: \(\cosh(x) = \frac{e^x + e^{-x}}{2}\)and\(\sinh(x) = \frac{e^x - e^{-x}}{2}\). These functions share some properties with trigonometric functions, such as addition formulas.

Step 2: Express in Terms of Exponential Functions

Using the definitions: - \(\cosh(x) + \sinh(x) = \frac{e^x + e^{-x}}{2} + \frac{e^x - e^{-x}}{2} = e^x\). Therefore, \((\cosh(x) + \sinh(x))^n = (e^x)^n = e^{nx}\).

Step 3: Use the Exponential Identities for Hyperbolic Functions

Find \(\cosh(nx)\) and \(\sinh(nx)\) using the definitions of hyperbolic functions:- \(\cosh(nx) = \frac{e^{nx} + e^{-nx}}{2}\)- \(\sinh(nx) = \frac{e^{nx} - e^{-nx}}{2}\).Thus, \(\cosh(nx) + \sinh(nx) = \frac{e^{nx} + e^{-nx}}{2} + \frac{e^{nx} - e^{-nx}}{2} = e^{nx}\).

Step 4: Equate the Expressions

Since both the expanded expression \((\cosh(x) + \sinh(x))^n = e^{nx}\) and the hyperbolic function identity \(\cosh(nx) + \sinh(nx) = e^{nx}\) are equivalent, the problem is proven true by matching the expressions: \((\cosh(x) + \sinh(x))^n = \cosh(nx) + \sinh(nx)\).

Key Concepts

cosh(x) definition

The hyperbolic cosine function, written as \(\cosh(x)\), is a mathematical function that comes into play in many calculus and algebra problems. It is defined using exponential functions as: \(\cosh(x) = \frac{e^x + e^{-x}}{2}\).
The \(\cosh(x)\) function is similar in spirit to the cosine function in trigonometry. However, instead of involving circular relationships, \(\cosh(x)\) relates to hyperbolas.

• The \(e^x\) term that forms part of its definition illustrates its growth.
• It is an even function, meaning \(\cosh(-x) = \cosh(x)\).
• It primarily deals with real numbers, providing positive outputs.

Understanding \(\cosh(x)\) is essential because it forms the backbone of various hyperbolic identities used in solving complex equations.

sinh(x) definition

The hyperbolic sine function, noted as \(\sinh(x)\), operates alongside \(\cosh(x)\) and follows a similar exponential pattern. Its definition is given by \(\sinh(x) = \frac{e^x - e^{-x}}{2}\).
Unlike \(\cosh(x)\), \(\sinh(x)\) is analogous to the sine function but pertains to hyperbolic function systems.

• The subtraction (\(-\)) in its formula causes its values to be zero-symmetric about the origin.
• \(\sinh(x)\) is an odd function, where \(\sinh(-x) = -\sinh(x)\).
• It represents the shape of a hyperbola, contrary to the circle-shaped relation in regular trigonometry.

Acquiring an understanding of \(\sinh(x)\) helps navigate through hyperbolic equations and simplifies handling of exponential laws.

hyperbolic identities

Hyperbolic identities are a set of math expressions involving hyperbolic functions like \(\sinh(x)\) and \(\cosh(x)\). These identities showcase relationships similar to trigonometric identities and are useful in various computations.
Some core hyperbolic identities include:

• Addition formula: \(\cosh(x+y) = \cosh(x)\cosh(y) + \sinh(x)\sinh(y)\)
• Subtraction formula: \(\cosh(x-y) = \cosh(x)\cosh(y) - \sinh(x)\sinh(y)\)
• Pythagorean identity: \(\cosh^2(x) - \sinh^2(x) = 1\)

These identities leverage the properties of exponential expressions within the hyperbolic functions, often facilitating easier integration and differentiation of mathematical functions in engineering and physics.
Mastering these formulas is paramount for resolving complex problems involving hyperbolic functions.

exponential functions in hyperbolic context

In the domain of hyperbolic functions, exponential functions are central components. Hyperbolic functions \(\cosh(x)\) and \(\sinh(x)\) are fundamentally based on exponential functions \(e^x\) and \(e^{-x}\).

• The addition and subtraction in \(\cosh(x)\) and \(\sinh(x)\) illustrate how exponential properties unfold.
• Exponential growth in these functions is often linear in appearance due to their mathematical formations.
• The expression and manipulation of these hyperbolic functions in exponential format offer substantial ease in calculus solving.

Understanding use of exponential powers simplifies and establishes solutions in expressions, making it especially beneficial when engaging with hyperbolic transformations in theoretical mathematics or real-world applications such as catenary curves in architecture.