Question

Find the numerical value of each expression

3.(a) \(\cosh \left( {\ln 5} \right)\) (b) \(\cosh 5\)

Short answer

a) The numerical value of \(\cosh \left( {\ln 5} \right)\) is \(\frac{{13}}{5}\).

b)The numerical value of \(\cosh 5\) is \(74.2099\).

Step-by-step solution

Definition of Hyperbolic function

The formulas for the hyperbolic function as shown below:

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

 Step 2: Evaluate the numerical value of the expression

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

\(\begin{aligned}\cosh \left( {\ln 5} \right) &= \frac{{{e^{\ln 5}} + {e^{ - \ln 5}}}}{2}\\ &= \frac{{5 + {{\left( {{e^{\ln 5}}} \right)}^{ - 1}}}}{2}\\ &= \frac{{5 + {5^{ - 1}}}}{2}\\ &= \frac{{5 + \frac{1}{5}}}{2}\end{aligned}\)

Solve further,

\(\begin{aligned}\cosh \left( {\ln 5} \right) &= \frac{{26}}{{10}}\\ &= \frac{{13}}{5}\end{aligned}\)

Thus, the numerical value of the expression is \(\frac{{13}}{5}\).

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

\(\begin{aligned}\cosh 5 &= \frac{{{e^5} + {e^{ - 5}}}}{2}\\ &= \frac{{148.4131 + 0.00673}}{2}\\ &= \frac{{148.4198}}{2}\\ \approx 74.2099\end{aligned}\)

Thus, the numerical value of the expression is \(74.2099\).