Problem 45 - Calculus Early Transcendentals: Pearson New International Edition

Question

Verify that the given equations are identities. \(\sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y\)

Step-by-Step Solution

Answer

The identity \( \sinh(x+y)=\sinh x \cosh y+\cosh x \sinh y \) is verified as true using hyperbolic function definitions.

Step 1: Understand the Definitions of Hyperbolic Functions

First, recall the definitions of the hyperbolic sine and cosine functions. The hyperbolic sine function is defined as \( \sinh(x) = \frac{e^x - e^{-x}}{2} \), and the hyperbolic cosine function is defined as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \). These definitions will be essential for verifying the identity.

Step 2: Expand \( \sinh(x+y) \) Using Hyperbolic Definitions

Using the definitions from Step 1, express \( \sinh(x+y) \):\[\sinh(x+y) = \frac{e^{x+y} - e^{-(x+y)}}{2} = \frac{e^x e^y - e^{-x} e^{-y}}{2}.\] This gives the expression of \( \sinh(x+y) \) in terms of exponentials.

Step 3: Expand the Right Side of the Given Identity

Next, substitute the hyperbolic function definitions into the right side of the identity \( \sinh x \cosh y + \cosh x \sinh y \):\[\sinh x \cosh y = \left(\frac{e^x - e^{-x}}{2}\right)\left(\frac{e^y + e^{-y}}{2}\right) = \frac{e^x e^y - e^x e^{-y} + e^{-x} e^y - e^{-x} e^{-y}}{4},\]\[\cosh x \sinh y = \left(\frac{e^x + e^{-x}}{2}\right)\left(\frac{e^y - e^{-y}}{2}\right) = \frac{e^x e^y - e^x e^{-y} + e^{-x} e^y - e^{-x} e^{-y}}{4}.\] Add these two results together.

Step 4: Combine and Simplify the Terms

Add the expanded results from Step 3:\[(\sinh x \cosh y + \cosh x \sinh y) = \frac{e^x e^y - e^x e^{-y} + e^{-x} e^y - e^{-x} e^{-y}}{2}.\] Notice that this expression is identical to the expanded \( \sinh(x+y) \) from Step 2, confirming the given identity is correct.

Key Concepts

Sinh Definition

The hyperbolic sine function, denoted as \( \sinh(x) \), is a fundamental concept in mathematics that appears frequently in calculus and complex analysis. Its definition is based on exponential functions, making it highly useful in various applications beyond typical trigonometry. The formal definition of the hyperbolic sine function is \( \sinh(x) = \frac{e^x - e^{-x}}{2} \). This equation tells you that the hyperbolic sine is derived from the combination of exponential growth and decay.

• \( e^x \) represents natural exponential growth as the exponent \( x \) increases.
• \( e^{-x} \) represents exponential decay as the exponent \( x \) becomes more negative.

As such, \( \sinh(x) \) takes on values based not only on\( x \) itself, but on how these exponential parts interact with one another. This unique approach distinguishes the hyperbolic sine from the circular sine function, which depends purely on circular motion.

Cosh Definition

The hyperbolic cosine function, or \( \cosh(x) \), complements the hyperbolic sine in many ways, offering symmetric behavior typical of hyperbolic functions. Its definition highlights how it mirrors \( \sinh(x) \) but through a different combination of exponential components. Formally, it is given by \( \cosh(x) = \frac{e^x + e^{-x}}{2} \).

• Here, both \( e^x \) and \( e^{-x} \) have positive contributions, emphasizing its symmetric nature.
• \( \cosh(x) \) is always positive, unlike \( \sinh(x) \), illustrating its distinct application in describing shapes like catenaries and hyperbolas.

Together with \( \sinh(x) \), the hyperbolic cosine helps express the relationship between exponential behavior and geometric constructs, lending to its utility in solving various mathematical problems involving hyperbolas and complex functions.

Hyperbolic Identity Verification

Verifying identities involving hyperbolic functions requires a solid understanding of their definitions and properties. In the given exercise, we verify the identity \( \sinh(x+y) = \sinh x \cosh y + \cosh x \sinh y \). This equality demonstrates how the summation and product of hyperbolic functions can be organized just like trigonometric identities.
By expressing \( \sinh(x+y) \) and combining the expressions derived from both sides' expansions, we see they are equivalent, affirming this identity. Such verifications are crucial because they establish foundational truths in mathematics that can be built upon in future studies.

• This verification not only solidifies our understanding of hyperbolic functions but also leads into more advanced topics, such as differential equations involving \( \sinh \) and \( \cosh \).
• Also, it underlines the symmetry and complementarity between hyperbolic sine and cosine, comparable in their role to the traditional circular sine and cosine.

Exponential Functions

At the heart of hyperbolic functions are the exponential functions \( e^x \) and \( e^{-x} \). These functions are foundational in mathematics, providing insight into growth and decay processes. The exponential function \( e^x \) signifies growth at a constant percentage rate, ubiquitous in natural and financial systems.

• The inverse, \( e^{-x} \), reflects decay—such as radioactive decay or cooling objects.
• Through the lens of hyperbolic functions, these exponentials reveal new perspectives by mimicking certain properties seen in their trigonometric counterparts.

Hyperbolic functions use these exponentials to express complex phenomena like hyperbolic geometry and relativistic physics. Their harmonic nature comes precisely from how exponential growth and decay flourish across both time and space. Understanding exponential functions is thus indispensable for capturing the finer points of hyperbolic behavior.