Question

Find the numerical value of each expression.

2.(a) \(\tanh 0\) (b) \(\tanh 1\)

Short answer

a) The numerical value of \(\tanh 0\) is 0.

b)The numerical value of \(\tanh 1\) is 0.76159.

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:

\(\begin{aligned}\tanh 0 &= \frac{{\sinh 0}}{{\cosh 0}}\\ &= \frac{{\frac{{{e^0} - {e^{ - 0}}}}{2}}}{{\frac{{{e^0} + {e^{ - 0}}}}{2}}}\\ &= \frac{{{e^0} - {e^{ - 0}}}}{{{e^0} + {e^{ - 0}}}}\\ &= 0\end{aligned}\)

Thus, the numerical value of \(\tanh 0\) is 0.

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

\(\begin{aligned}\tanh 1 &= \frac{{\sinh 1}}{{\cosh 1}}\\ &= \frac{{\frac{{{e^1} - {e^{ - 1}}}}{2}}}{{\frac{{{e^1} + {e^{ - 1}}}}{2}}}\\ &= \frac{{{e^1} - {e^{ - 1}}}}{{{e^1} + {e^{ - 1}}}}\\ &= \frac{{{e^2} - 1}}{{{e^2} + 1}}\end{aligned}\)

Solve further,

\(\begin{aligned}\tanh 1 &= \frac{{6.3890}}{{8.38905}}\\ \approx 0.76159\end{aligned}\)

Thus, the numerical value of \(\tanh 1\) is 0.76159.