Expressing 9 Sinh(x) + 4 Cosh(x) In Terms Of E^x And E^-x A Step-by-Step Guide

Jul 9, 2025 by qnaftunila 79 views

In the realm of mathematics, particularly in calculus and complex analysis, hyperbolic functions play a crucial role. Hyperbolic functions, such as sinh(x) and cosh(x), are counterparts to the trigonometric functions, but instead of circles, they are based on hyperbolas. These functions often appear in various applications, ranging from physics to engineering. Expressing these hyperbolic functions in terms of exponential functions, specifically e^x and e^-x, provides a fundamental way to understand their behavior and simplifies many calculations. This article delves into how to express the linear combination 9 sinh(x) + 4 cosh(x) using exponential terms, offering a detailed, step-by-step explanation and highlighting the underlying mathematical principles. This exploration is beneficial for students, educators, and professionals who need a solid grasp of these transformations. By the end of this article, you will clearly understand how to manipulate hyperbolic functions and their exponential equivalents, which will enhance your problem-solving skills in more advanced mathematical contexts.

Understanding Hyperbolic Functions

Before we dive into the specific expression, let's establish a clear understanding of the hyperbolic functions sinh(x) and cosh(x). Sinh(x), pronounced "sinch x," stands for the hyperbolic sine of x, while cosh(x), pronounced "cosh x," represents the hyperbolic cosine of x. These functions are defined in terms of exponential functions as follows:

sinh(x) = (e^x - e^-x) / 2
cosh(x) = (e^x + e^-x) / 2

These definitions are pivotal in transforming hyperbolic expressions into exponential ones. The beauty of these exponential forms lies in their simplicity and the ease with which they can be manipulated using algebraic techniques. Understanding these definitions is essential for anyone delving into the world of hyperbolic functions, as they form the building blocks for more complex operations and applications. The elegance of expressing hyperbolic functions through exponentials is not just a mathematical curiosity; it has practical implications in various fields, allowing for streamlined calculations and a deeper understanding of the functions' properties. This foundation is crucial as we move forward to tackle the specific problem of expressing 9 sinh(x) + 4 cosh(x) in exponential terms.

Expressing 9 sinh(x) + 4 cosh(x) in Exponential Terms

Now, let's tackle the main task: expressing 9 sinh(x) + 4 cosh(x) in terms of e^x and e^-x. To accomplish this, we will use the definitions of sinh(x) and cosh(x) in terms of exponential functions. We will substitute these definitions into the given expression and then simplify the result. This process not only allows us to rewrite the expression but also deepens our understanding of how linear combinations of hyperbolic functions can be transformed into exponential forms. The step-by-step approach ensures clarity and allows for a thorough grasp of the methodology, making it easier to apply similar techniques to other problems involving hyperbolic and exponential functions. This section will serve as a practical guide to performing such transformations, a skill that is highly valuable in various mathematical and scientific applications.

Step-by-Step Transformation

Substitute the Definitions:

We start by substituting the exponential definitions of sinh(x) and cosh(x) into the expression 9 sinh(x) + 4 cosh(x):

9 sinh(x) + 4 cosh(x) = 9 * ((e^x - e^-x) / 2) + 4 * ((e^x + e^-x) / 2)

This substitution is the foundational step in our transformation process. It directly applies the definitions we established earlier, setting the stage for further simplification. By replacing the hyperbolic functions with their exponential equivalents, we shift the problem from the domain of hyperbolic functions to the more familiar territory of exponential functions. This substitution is a powerful technique that simplifies the expression and makes it easier to manipulate algebraically. It is a common practice in mathematics to transform expressions into a more manageable form, and this step exemplifies that approach.
Distribute and Simplify:

Next, we distribute the constants 9 and 4 across their respective terms and simplify the expression:

9 * ((e^x - e^-x) / 2) + 4 * ((e^x + e^-x) / 2) = (9e^x - 9e^-x) / 2 + (4e^x + 4e^-x) / 2

This step involves basic arithmetic operations but is crucial for organizing the terms and preparing them for combination. By distributing the constants, we eliminate the parentheses and make the individual terms more accessible. The simplification process is a key aspect of mathematical problem-solving, allowing us to reduce complex expressions into simpler, more manageable forms. This distribution is not just a mechanical operation; it is a strategic move that sets the expression up for the next phase of simplification, which involves combining like terms.
Combine Like Terms:

Now, we combine the terms with e^x and e^-x:

(9e^x - 9e^-x) / 2 + (4e^x + 4e^-x) / 2 = (9e^x + 4e^x) / 2 + (-9e^-x + 4e^-x) / 2

Combining like terms is a fundamental algebraic technique that simplifies expressions by grouping similar elements together. In this case, we are grouping the e^x terms and the e^-x terms separately. This step is essential for condensing the expression and making it more readable. By combining like terms, we reduce the number of individual terms in the expression, which makes it easier to analyze and work with. This process is a standard practice in mathematical simplification and is a critical skill for anyone working with algebraic expressions.
Final Simplification:

Finally, we add the coefficients and present the expression in its simplest form:

(9e^x + 4e^x) / 2 + (-9e^-x + 4e^-x) / 2 = (13e^x) / 2 + (-5e^-x) / 2

This final simplification step completes the transformation, giving us a clear and concise expression in terms of e^x and e^-x. By performing the additions, we arrive at the most reduced form of the expression, which is easier to interpret and use in further calculations. This final step is the culmination of the entire process, demonstrating the power of algebraic manipulation in simplifying complex expressions. The simplified form not only provides a clear answer to the original problem but also highlights the underlying structure of the expression, making it easier to understand the relationship between hyperbolic and exponential functions.

Therefore, 9 sinh(x) + 4 cosh(x) can be expressed as:

(13/2)e^x + (-5/2)e^-x

This result clearly shows the transformation of the hyperbolic expression into its exponential equivalent, fulfilling the objective of the problem.

Result and Discussion

After performing the step-by-step transformation, we have successfully expressed 9 sinh(x) + 4 cosh(x) in terms of e^x and e^-x. The final expression is:

(13/2)e^x + (-5/2)e^-x

This result showcases the linear combination of exponential terms that is equivalent to the original hyperbolic expression. The transformation process involved substituting the exponential definitions of sinh(x) and cosh(x), distributing constants, combining like terms, and simplifying the result. Each step was crucial in achieving the final expression, highlighting the importance of careful algebraic manipulation. The resulting expression not only answers the specific question but also provides a deeper understanding of the relationship between hyperbolic and exponential functions. This equivalence is a fundamental concept in mathematics and has significant implications in various fields, including calculus, differential equations, and complex analysis.

Significance of the Transformation

The ability to express hyperbolic functions in terms of exponential functions is significant for several reasons. Firstly, it simplifies many calculus operations. Differentiation and integration of exponential functions are often more straightforward than those of hyperbolic functions. This makes it easier to solve differential equations and analyze the behavior of various mathematical models. Secondly, this transformation provides insights into the properties of hyperbolic functions. By understanding their exponential forms, we can better grasp their symmetry, asymptotic behavior, and relationships with other functions. For instance, the exponential form reveals that sinh(x) is an odd function and cosh(x) is an even function, properties that are not immediately apparent from their hyperbolic definitions alone. Finally, expressing hyperbolic functions in exponential terms is essential in complex analysis. Complex exponentials play a central role in the study of complex functions, and the connection between hyperbolic and exponential functions is crucial for extending real analysis concepts to the complex plane.

Conclusion

In conclusion, we have demonstrated how to express 9 sinh(x) + 4 cosh(x) in terms of e^x and e^-x. The step-by-step process involved substituting the exponential definitions of sinh(x) and cosh(x), distributing constants, combining like terms, and simplifying the result. The final expression, (13/2)e^x + (-5/2)e^-x, clearly represents the transformation of the hyperbolic expression into its exponential equivalent. This exercise underscores the importance of understanding the fundamental definitions and properties of hyperbolic and exponential functions.

The ability to transform between hyperbolic and exponential forms is a valuable skill in mathematics and its applications. It simplifies calculations, provides deeper insights into the behavior of functions, and bridges the gap between real and complex analysis. This article has provided a comprehensive guide to performing such transformations, equipping readers with the knowledge and tools to tackle similar problems. Whether you are a student, educator, or professional, mastering these techniques will undoubtedly enhance your mathematical proficiency and problem-solving capabilities. The connections between different mathematical concepts, such as hyperbolic and exponential functions, are what make mathematics a powerful and cohesive discipline. By exploring these connections, we gain a richer understanding of the mathematical world and its applications in the real world.