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Section The Hyperbolic Functions

Similar to the trigonometric functions in Euclidean trigonometry, the following functions are important in hyperbolic trigonometry. These occur in engineering problems because they appear as solutions to differential equations modeling electrical circuits and they can be used to represent the curve that results when hanging a wire between two poles (think telephone lines).

Definition 5.34.

The hyperbolic sine and hyperbolic cosine functions are denoted by sinh and cosh and defined by:

\begin{equation*} \sinh(x) = \frac{e^x - e^{-x}}{2} \mbox{ and } \cosh(x) = \frac{e^x +e^{-x}}{2}. \end{equation*}

The hyperbolic tangent, hyperbolic cotangent, hyperbolic secant, and hyperbolic cosecant functions are denoted by tanh, coth, sech, csch and defined by:

\begin{equation*} \dsp \tanh(x) = \frac{\sinh(x)}{\cosh(x)}, \;\; \coth(x) = \frac{\cosh(x)}{\sinh(x)}, \;\; \dsp \sech(x) = \frac{1}{\cosh(x)}, \;\; \mbox{ and } \;\; \dsp \csch(x) = \frac{1}{\sinh(x)}. \end{equation*}
Problem 5.35.

Assume that \(x\) is a number and show that \(\cosh^2(x) - \sinh^2(x) = 1\text{.}\)

Problem 5.36.

Assume that \(x\) is a number and show that each of the following is true.

  1. \(\dsp \frac{d}{dx} \sinh(x) = \cosh(x)\)

  2. \(\dsp \frac{d}{dx} \tanh(x) = \sech^2(x)\)

While many inverse functions (such as \(invtan\)) cannot be written out explicitly as a function of \(x\) only, all of the inverse hyperbolic trigonometric functions can be.

Problem 5.37.

Rewrite \(\invcosh\) as a function of \(x\) and without using any hyperbolic functions by solving \(\dsp y = \frac{e^x+e^{-x}}{2}\) for \(x.\)

Problem 5.38.

Show that \(\dsp \frac{d}{dx} \invsinh (x)=\frac{1}{\sqrt{x^2 +1}}\) by letting \(y = \invsinh(x),\) applying implicit differentiation to \(\sinh(y) = x,\) and using the identity \(\cosh^2(x) - \sinh^2 (x) = 1\text{.}\)

Problem 5.39.

Show that \(\dsp \frac{d}{dx} \invsech (x)= -\frac{1}{x \sqrt{1-x^2}}\) for \(0 \lt x \leq 1.\)

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