Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.


d. sech (sinh 0)

Verified step by step guidance

1. Recall the definitions of the hyperbolic functions involved: \( \sinh x = \frac{e^{x} - e^{-x}}{2} \) and \( \sech x = \frac{1}{\cosh x} \), where \( \cosh x = \frac{e^{x} + e^{-x}}{2} \).
2. First, evaluate \( \sinh 0 \) by substituting \( x = 0 \) into the definition: \( \sinh 0 = \frac{e^{0} - e^{0}}{2} \).
3. Simplify the expression for \( \sinh 0 \) to find its exact value.
4. Next, substitute the value of \( \sinh 0 \) into the expression \( \sech(\sinh 0) \), which becomes \( \sech(\text{value}) = \frac{1}{\cosh(\text{value})} \).
5. Finally, evaluate \( \cosh(\text{value}) \) using its definition and simplify to find \( \sech(\sinh 0) \).

Key Concepts

Hyperbolic Sine Function (sinh)

The hyperbolic sine function, sinh(x), is defined as (e^x - e^(-x))/2. It is an odd function and maps real numbers to real numbers. Evaluating sinh at zero gives sinh(0) = 0, which is a key step in simplifying expressions involving hyperbolic functions.

Hyperbolic Secant Function (sech)

The hyperbolic secant function, sech(x), is defined as 1/cosh(x), where cosh(x) = (e^x + e^(-x))/2. It is always positive for real x and is used to find the reciprocal of the hyperbolic cosine. Understanding sech helps in evaluating expressions like sech(sinh 0).

Function Composition and Simplification

Function composition involves applying one function to the result of another, such as sech(sinh 0). Simplifying requires evaluating the inner function first, then applying the outer function. This stepwise approach is essential for correctly simplifying nested hyperbolic expressions.