Problem 383 - Calculus Volume 1

Question

Prove \(\sinh (x+y)=\sinh (x) \cosh (y)+\cosh (x) \sinh (y)\) changing the expression to exponentials.

Step-by-Step Solution

Answer

The identity \( \sinh(x+y) = \sinh(x) \cosh(y) + \cosh(x) \sinh(y) \) is proven by converting to exponentials and simplifying both sides.

Step 1: Recall Hyperbolic Sine Definition

The hyperbolic sine function is defined as \( \sinh(x) = \frac{e^x - e^{-x}}{2} \). This will be important for converting all hyperbolic functions into exponential form.

Step 2: Recall Hyperbolic Cosine Definition

The hyperbolic cosine function is defined as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \). Like the hyperbolic sine, this will allow us to convert the expression into exponential form.

Step 3: Express \( \sinh(x+y) \) using Exponentials

Use the definitions of hyperbolic functions to express \( \sinh(x+y) \) as follows: \[ \sinh(x+y) = \frac{e^{x+y} - e^{-(x+y)}}{2}. \]

Step 4: Expand \( e^{x+y} \) and \( e^{-(x+y)} \)

Expand the exponential expressions: \( e^{x+y} = e^x \cdot e^y \) and \( e^{-(x+y)} = e^{-x} \cdot e^{-y} \). Substitute these into the expression for \( \sinh(x+y) \): \[ \sinh(x+y) = \frac{e^x e^y - e^{-x} e^{-y}}{2}. \]

Step 5: Express \( \sinh(x) \cosh(y) \) and \( \cosh(x) \sinh(y) \) using Exponentials

Substitute the exponential forms of hyperbolic functions: \[ \sinh(x) \cosh(y) = \frac{(e^x - e^{-x})(e^y + e^{-y})}{4} = \frac{e^{x+y} - e^{x-y} + e^{-x+y} - e^{-(x+y)}}{4}, \] \[ \cosh(x) \sinh(y) = \frac{(e^x + e^{-x})(e^y - e^{-y})}{4} = \frac{e^{x+y} + e^{x-y} - e^{-x+y} - e^{-(x+y)}}{4}. \]

Step 6: Combine and Simplify the Right Side Expression

Combine the expressions from Step 5: \[ \sinh(x) \cosh(y) + \cosh(x) \sinh(y) = \frac{e^{x+y} - e^{x-y} + e^{-x+y} - e^{-(x+y)} + e^{x+y} + e^{x-y} - e^{-x+y} - e^{-(x+y)}}{4}. \]Simplify the terms: \[ \sinh(x) \cosh(y) + \cosh(x) \sinh(y) = \frac{2e^{x+y} - 2e^{-(x+y)}}{4} = \frac{e^{x+y} - e^{-(x+y)}}{2}. \]

Step 7: Conclude the Proof

Since both expressions for \( \sinh(x+y) \) from Steps 3 and 6 are equivalent, \( \sinh(x+y) = \sinh(x) \cosh(y) + \cosh(x) \sinh(y) \) is proven true using exponential definitions.

Key Concepts

Hyperbolic Sine

The hyperbolic sine function, commonly denoted as \( \sinh(x) \), is an important hyperbolic function in mathematics. Calculation-wise, it relates closely to exponential functions. The hyperbolic sine is defined as:

• \( \sinh(x) = \frac{e^x - e^{-x}}{2} \)

The hyperbolic sine essentially measures a type of 'height' or 'component' similar to how sine measures height in circular trigonometry. Unlike traditional trigonometric functions which relate to circular forms, hyperbolic functions relate more to hyperbolas.
Understanding \( \sinh(x) \) through its exponential form allows us to seamlessly interchange between exponential and hyperbolic expressions, which is crucial in simplifying and proving mathematical relationships. By using this exponential form, one can solve complex equations involving hyperbolic functions with the ease of manipulating exponentials.

Hyperbolic Cosine

The hyperbolic cosine, denoted as \( \cosh(x) \), is the complementary function to \( \sinh(x) \). It provides a kind of 'width' measurement, akin to cosine giving width in circular trigonometry. Its definition through exponential functions is:

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

Similar to \( \sinh(x) \), expressing \( \cosh(x) \) in terms of exponentials simplifies the analysis and manipulation of these functions. This is particularly useful when proving identities or transforming equations involving hyperbolic functions.
Using exponentials to express \( \cosh(x) \) harmonizes hyperbolic equations and reveals deep geometric properties that parallel those of circular trigonometry. This relationship underlies many proofs and solutions in higher mathematics, including calculus and hyperbolic geometry.

Exponential Functions

Exponential functions form the backbone of concepts in calculus and complex analysis. Typically expressed as \( e^x \), these functions exhibit rapid growth and are fundamental in describing everything from population growth models to compound interest calculations.

• The base of the natural logarithm \( e \), approximately 2.718, is used because of its unique properties in calculus.
• Exponentials can either grow swiftly with positive exponents or diminish to zero with negative exponents.
• They are also periodic when extended into complex numbers, showing fascinating behavior in both real and complex planes.

Hyperbolic functions like \( \sinh(x) \) and \( \cosh(x) \) demonstrate how exponentials can create non-intuitive yet highly significant functions. Understanding exponentials allows one to manipulate mathematical expressions, facilitating proofs and simplifying complex calculations which involve growth or decay patterns. Hyperbolic functions are, essentially, particular combinations of exponential growth and decay.

Proof in Mathematics

Proving mathematical statements involves establishing truth through logical reasoning and previously established premises. The process of proof solidifies theoretical understanding and supports the correctness of mathematical identities.

• Proofs often start by stating known identities and concepts, like the hyperbolic sine and cosine in terms of exponentials.
• The transformation and simplification of equations help in illustrating the relationship between the original problem and the statement to be proved.
• Combining various mathematical rules like algebraic manipulation or substituting equivalent expressions is essential.

In the proof of hyperbolic identities, changing expressions to exponentials often reveals the underlying structure of the problem. In the case of proving \( \sinh(x+y) = \sinh(x) \cosh(y) + \cosh(x) \sinh(y) \), using exponential definitions aided in breaking down complex functions into manageable parts, aligning them naturally to demonstrate equality. Proofs thereby offer clarity and validate mathematical intuition, serving as the foundation upon which more complex ideas can be constructed.