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


a. cosh 0

Verified step by step guidance

1. Recall the definition of the hyperbolic cosine function: \(\cosh x = \frac{e^{x} + e^{-x}}{2}\).
2. Substitute \(x = 0\) into the definition: \(\cosh 0 = \frac{e^{0} + e^{-0}}{2}\).
3. Evaluate the exponentials: \(e^{0} = 1\) and \(e^{-0} = 1\).
4. Add the values in the numerator: \$1 + 1 = 2$.
5. Divide by 2 to simplify: \(\cosh 0 = \frac{2}{2}\).

Key Concepts

Definition of Hyperbolic Cosine (cosh)

The hyperbolic cosine function, cosh(x), is defined as (e^x + e^(-x)) / 2. It is an even function and describes the shape of a hanging cable or chain (catenary). Understanding this definition allows direct evaluation of cosh at any point, including zero.

Properties of Exponential Functions

Exponential functions e^x and e^(-x) are fundamental in defining hyperbolic functions. Knowing that e^0 = 1 simplifies calculations, especially when evaluating cosh(0), since it involves e^0 and e^(-0). This property helps in simplifying expressions without a calculator.

Evaluating Functions at Specific Points

Evaluating a function at a specific point means substituting the value into the function's formula and simplifying. For cosh(0), substituting x=0 into the definition and simplifying using known values of exponentials yields the exact result without approximation.