Question

Find the numerical value of each expression.

6.(a) \(\sinh 1\) (b) \({\sinh ^{ - 1}}1\)

Short answer

a) The numerical value of \(\sinh 1\) is \(1.17520\).

b)The numerical value of \({\sinh ^{ - 1}}1\) is \(0.88137\).

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:

\(\begin{aligned}\sinh 1 &= \frac{{{e^1} - {e^{ - 1}}}}{2}\\ &= \frac{{2.71828 - 0.36787}}{2}\\ &= \frac{{2.3504}}{2}\\ &\approx 1.17520\end{aligned}\)

Thus, the numerical value of the expression is \(1.17520\).

b)Use the above formula to obtain the numerical value of the expression as shown below:

\(\begin{aligned}{\sinh ^{ - 1}}1 &= \ln \left( {1 + \sqrt {{1^2} + 1} } \right)\,\,\,\,\,\,\,\,\,\left( {{\mathop{\rm since}\nolimits} \,\,\,{{\sinh }^{ - 1}}x = \ln \left( {x + \sqrt {{x^2} + 1} } \right)} \right)\\ &= \ln \left( {1 + \sqrt 2 } \right)\\ &= \ln \left( {2.4142} \right)\\ &\approx 0.88137\end{aligned}\)

Thus, the numerical value of the expression is\(0.88137\)

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