Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.
f. sinh (2 ln 3)

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

Substitute \(x = 2 \ln 3\) into the definition: \(\sinh(2 \ln 3) = \frac{e^{2 \ln 3} - e^{-2 \ln 3}}{2}\).

Use the property of exponents and logarithms: \(e^{a \ln b} = b^{a}\). So, \(e^{2 \ln 3} = 3^{2}\) and \(e^{-2 \ln 3} = 3^{-2}\).

Rewrite the expression using these simplifications: \(\sinh(2 \ln 3) = \frac{3^{2} - 3^{-2}}{2}\).

Simplify the powers: \(3^{2} = 9\) and \(3^{-2} = \frac{1}{9}\). So, \(\sinh(2 \ln 3) = \frac{9 - \frac{1}{9}}{2}\).

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Definition of Hyperbolic Sine Function (sinh)

The hyperbolic sine function, sinh(x), is defined as (e^x - e^(-x)) / 2. It is analogous to the sine function but based on exponential functions, which allows simplification when the input is expressed in terms of logarithms or exponentials.

Properties of Logarithms and Exponentials

The natural logarithm ln(x) and the exponential function e^x are inverse functions. Using properties like e^(ln a) = a helps simplify expressions involving compositions of exponentials and logarithms, which is essential for evaluating sinh(2 ln 3).

Double Angle Formula for Hyperbolic Sine

The double angle formula for sinh states that sinh(2x) = 2 sinh(x) cosh(x). This identity allows breaking down sinh(2 ln 3) into simpler parts involving sinh(ln 3) and cosh(ln 3), which can then be evaluated using exponential definitions.

Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.f. sinh (2 ln 3)