Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.
g. cosh² 1

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Recall the definition of the hyperbolic cosine function: \(\cosh x = \frac{e^{x} + e^{-x}}{2}\).

Express \(\cosh^2 1\) as \(\left( \cosh 1 \right)^2 = \left( \frac{e^{1} + e^{-1}}{2} \right)^2\).

Square the expression inside the parentheses: \(\left( \frac{e^{1} + e^{-1}}{2} \right)^2 = \frac{(e^{1} + e^{-1})^2}{4}\).

Expand the numerator using the formula \((a + b)^2 = a^2 + 2ab + b^2\): \((e^{1})^2 + 2 e^{1} e^{-1} + (e^{-1})^2 = e^{2} + 2 + e^{-2}\).

Combine the results to write \(\cosh^2 1 = \frac{e^{2} + 2 + e^{-2}}{4}\), which is the simplified exact expression.

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주요 개념

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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 is analogous to the cosine function in trigonometry but based on exponential functions. Understanding this definition allows direct evaluation of cosh at any real number.

Squaring Hyperbolic Functions

Squaring cosh(x) means computing (cosh(x))^2, which can be expressed using exponential terms or simplified using hyperbolic identities. Recognizing how to handle powers of hyperbolic functions is essential for simplification and evaluation.

Hyperbolic Identity: cosh²(x) - sinh²(x) = 1

This fundamental identity relates cosh and sinh functions, similar to the Pythagorean identity in trigonometry. It can be used to rewrite or simplify expressions involving cosh²(x), especially when combined with expressions involving sinh(x).

Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.g. cosh² 1