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Chapter 7: Problem 11

Identities Prove each identity using the definitions of the hyperbolic functions. \(\tanh x=\frac{e^{2 x}-1}{e^{2 x}+1}\)

Short Answer

Expert verified
Question: Prove that the hyperbolic tangent function, \(\tanh x\), can be represented as \(\tanh x = \frac{e^{2x} - 1}{e^{2x} + 1}\) using the definitions of the hyperbolic sine and cosine functions. Answer: By substituting the definitions of the hyperbolic sine and cosine functions into the expression for the hyperbolic tangent function and simplifying, we can prove that \(\tanh x = \frac{e^{2x} - 1}{e^{2x} + 1}\).

Step by step solution

01

Write the definition of the hyperbolic tangent

The hyperbolic tangent function is defined as: \(\tanh x = \frac{\sinh x}{\cosh x}\)
02

Substitute the definitions of the hyperbolic sine and cosine functions

Using the definitions of the hyperbolic sine and cosine functions, we have: \(\tanh x = \frac{\frac{e^x - e^{-x}}{2}}{\frac{e^x + e^{-x}}{2}}\)
03

Simplify the fraction

To simplify the fraction, we can multiply the numerator and denominator by 2: \(\tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}}\)
04

Multiply the numerator and denominator by \(e^x\)

To get rid of the negative exponent, multiply the numerator and denominator by \(e^x\): \(\tanh x = \frac{(e^x - e^{-x})(e^x)}{(e^x + e^{-x})(e^x)}\)
05

Expand the expressions

Now, expand the expressions in the numerator and denominator: \(\tanh x = \frac{e^{2x} - 1}{e^{2x} + e^x}\)
06

Factor out \(e^x\) from the denominator

Rewrite the denominator to more clearly show the factor of \(e^x\): \(\tanh x = \frac{e^{2x} - 1}{e^x (e^x + 1)}\)
07

Simplify the fraction

Finally, since we multiplied both the numerator and denominator by \(e^x\) in step 4, we can simplify the fraction again to obtain the final result: \(\tanh x = \frac{e^{2x} - 1}{e^{2x} + 1}\) The given identity has been proven using the definitions of the hyperbolic functions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hyperbolic Tangent
The hyperbolic tangent function is an essential hyperbolic function with the symbol \( \tanh x \). In terms of math, it is defined as the ratio of the hyperbolic sine to the hyperbolic cosine. Here's the formula:
  • \( \tanh x = \frac{\sinh x}{\cosh x} \)
This function often appears in various mathematical contexts, similar to how the tangent function works in trigonometry. It `measures` how the hyperbolic sine balances with the hyperbolic cosine, giving a value between -1 and 1.
Students can imagine this function as a smooth transition, unlike the abrupt jumps you see in trigonometric tangent functions. Understanding \( \tanh x \) can provide insights into modeling growth patterns or in areas involving rapid changes.
Hyperbolic Sine
Understanding the hyperbolic sine function \( \sinh x \) is another step in grasping hyperbolic functions. It is a mathematical function similar to the usual sine but reflects hyperbolic geometry. Here's the definition:
  • \( \sinh x = \frac{e^x - e^{-x}}{2} \)
Composable and smooth, \( \sinh x \) produces a wave-like curve that can describe many natural processes and phenomena.
One of the main applications of the hyperbolic sine function is in physics, particularly in calculations involving the theory of relativity and hyperbolic arches. Its ease of calculation using its exponential form makes it a favorite in mathematical modeling.
Hyperbolic Cosine
The hyperbolic cosine function \( \cosh x \) plays an important role in analyzing curves and shapes in the hyperbolic context. Defined much like its trigonometric cousin but rooted in hyperbolic terms:
  • \( \cosh x = \frac{e^x + e^{-x}}{2} \)
While \( \sinh x \) captures the asymmetry, \( \cosh x \) is all about balance, representing evenness and reaching its minimum at zero.
This function has applications in various branches of science and engineering, such as solving differential equations that model real-world phenomena.
Beyond this, its property of having a rapid rise helps understand insulation and growth issues, shielding structures from harsh conditions.
Algebraic Simplification
Algebraic simplification is a powerful tool in mathematics that reduces complications in expressions. Here, it helps simplify expressions involving hyperbolic functions.
For example, during the simplification of the hyperbolic tangent function, arithmetic operations like multiplying by \( e^x \) are used to eliminate negative exponents. Here's why this is important:
  • It makes equations more manageable and easier to interpret.
  • Simplified forms of expressions are often easier to differentiate and integrate.
  • Helps maintain consistency across solutions.
Embrace the idea that simplification is half art, half technique. It allows students to navigate equations with confidence, choosing the best form to achieve clarity and precision.

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