Problem 92 - Precalculus

Question

In calculus the following two functions are studied: $$ \sinh x=\frac{e^{x}-e^{-x}}{2} \quad \text { and } \quad \cosh x=\frac{e^{x}+e^{-x}}{2} $$ $$\text { Show that } \cosh x+\sinh x=e^{x}$$

Step-by-Step Solution

Answer

\( \cosh x + \sinh x = e^{x} \) is proven by combining terms and simplifying the fraction.

Step 1: Write Down Given Equations

We are given two hyperbolic functions: \( \sinh x = \frac{e^{x} - e^{-x}}{2} \) and \( \cosh x = \frac{e^{x} + e^{-x}}{2} \). We are required to prove that \( \cosh x + \sinh x = e^{x} \).

Step 2: Express the Sum

Add the two given functions: \( \cosh x + \sinh x = \left( \frac{e^{x} + e^{-x}}{2} \right) + \left( \frac{e^{x} - e^{-x}}{2} \right) \).

Step 3: Simplify the Expression

Combine the fractions: \( \cosh x + \sinh x = \frac{(e^{x} + e^{-x}) + (e^{x} - e^{-x})}{2} \).

Step 4: Combine Terms

Notice that the \(e^{-x}\) terms cancel each other out: \( \cosh x + \sinh x = \frac{2e^{x}}{2} \).

Step 5: Simplify the Fraction

Simplify the expression to obtain: \( \cosh x + \sinh x = e^{x} \).

Step 6: Conclusion

We have shown that the expression \( \cosh x + \sinh x \) simplifies to \( e^{x} \), thus proving the required result.

Key Concepts

Understanding the Sinh Function

The sinh function, or hyperbolic sine function, is a core concept in calculus and mathematical analysis. It looks similar to the familiar sine function from trigonometry but is defined differently.
Instead of being related to circular functions, the sinh function is based on exponential functions, which involve Euler's number, \( e \). The sinh function is expressed as:

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

This formula uses the exponential growth and decay captured by \( e^x \) and \( e^{-x} \).
The sinh function can be interpreted as representing the shape of a hanging cable or chain, known as a catenary. This feature makes it useful in physics and engineering applications that involve hyperbolic geometries.
Additionally, the sinh function benefits from several neat properties: it is an odd function, meaning \( \sinh(-x) = -\sinh(x) \), and it is continuous and has a smooth curve throughout its domain.

Exploring the Cosh Function

The cosh function, or hyperbolic cosine function, is another foundational hyperbolic function in calculus. It mirrors the behavior of the cosine function but is based on exponentially related behavior.
The standard expression for the cosh function is:

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

The cosh function shows the mean of exponential growth and decay, emphasized through the combination of \( e^x \) and \( e^{-x} \).
Graphically, the cosh function resembles a U-shaped curve, standing above the x-axis, suggesting a minimum point at \( x = 0 \).
In terms of properties, it's essential to note that the cosh function is an even function, which means \( \cosh(-x) = \cosh(x) \), making it symmetric about the y-axis.
This function proves to be significant in various applications like modeling the shape of a hanging cable, as a catenary curve, and describing hyperbolic as well as hyperbolic geometries.

Understanding Proofs in Mathematics

Proofs are vital in mathematics. They provide a rigorous way of showing that a statement or theorem is universally true. It offers certainty and solidification to mathematical discoveries and assertions.
There are various types of proofs:

• Direct Proof: Starts with known facts and uses logical steps to arrive at the statement to be proven.
• Indirect Proof: Includes methods like proof by contradiction, where the opposite of the statement is assumed false and demonstrates a contradiction.
• Inductive Proof: Shows a statement is true for a base case and assumes it's true for \( n \), proving it for \( n+1 \).

In the given problem, to prove that \( \cosh x + \sinh x = e^x \), we use a direct proof method.
This method involves manipulation of equations step-by-step to illustrate that considering the formulas for \( \cosh x \) and \( \sinh x \), the combination simplifies exactly to \( e^x \).

The Role of Calculus

Calculus is a branch of mathematics that studies continuous change, much like geometry is the study of shapes and algebra the study of operations. Calculus is widely involved in various domains of mathematics and science.
It comprises two fundamental aspects:

• Differential Calculus: Focuses on the concept of a derivative, which represents a rate of change.
• Integral Calculus: Concerns itself with the concept of integration, allowing the determination of the total size or value, such as area under a curve.

When examining hyperbolic functions such as \( \sinh x \) and \( \cosh x \), calculus plays a key role in understanding their behavior.
It helps to study the derivatives and integrals of these functions, facilitating greater insight into their properties and enabling solutions to real-world problems, which require modeling and optimization solutions.
For instance, in calculus, finding the derivative of \( \sinh x \) is straightforward: it leads to \( \cosh x \), demonstrating the close relation to trigonometric functions in terms of derivative relationships.