Question

Find the numerical value of each expression

4.(a) \(\sinh 4\) (b) \(\sinh \left( {\ln 4} \right)\)

Short answer

a) The numerical value of \(\sinh 4\) is\(27.28992\).

b)The numerical value of \(\sinh \left( {\ln 4} \right)\) is \(\frac{{15}}{8}\).

Step-by-step solution

Definition of Hyperbolic function

The formulas for the hyperbolic function as shown below:

\(\begin{aligned}\sinh x = \frac{{{e^x} - {e^{ - x}}}}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\mathop{\rm csch}\nolimits} x = \frac{1}{{\sinh x}}\\\cosh x = \frac{{{e^x} + {e^{ - x}}}}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,{\mathop{\rm sech}\nolimits} x = \frac{1}{{\cosh x}}\\\tanh x = \frac{{\sinh x}}{{\cosh x}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\coth x = \frac{{\cosh x}}{{\sinh x}}\end{aligned}\)

Evaluate the numerical value of the expression

a)Evaluate the numerical value of the expression as shown below:

\(\begin{aligned}\sinh 4 &= \frac{{{e^4} - {e^{ - 4}}}}{2}\\ &= \frac{{54.59815 - 0.01831}}{2}\\ &= \frac{{54.5798}}{2}\\ \approx 27.28992\end{aligned}\)

Thus, the numerical value of the expression is \(27.28992\).

b)Evaluate the numerical value of the expression as shown below:

\(\begin{aligned}\sinh \left( {\ln 4} \right) &= \frac{{{e^{\ln 4}} - {e^{ - \ln 4}}}}{2}\\ &= \frac{{4 - {{\left( {{e^{\ln 4}}} \right)}^{ - 1}}}}{2}\\ &= \frac{{4 - {4^{ - 1}}}}{2}\\ &= \frac{{4 - \frac{1}{4}}}{2}\\ &= \frac{{15}}{8}\end{aligned}\)

Thus, the numerical value of the expression is \(\frac{{15}}{8}\).