3.6 Hyperbolic Functions

1 Define the Hyperbolic Functions

Functions involving certain combinations of \(e^{x}\) and \(e^{-x}\) occur so frequently in applied mathematics that they warrant special study. These functions, called hyperbolic functions, have properties similar to those of trigonometric functions. Because of this, they are named the hyperbolic sine (\(\sinh \)), the hyperbolic cosine (\(\cosh \)), the hyperbolic tangent (\(\tanh \)), the hyperbolic cotangent (\(\coth \)), the hyperbolic cosecant (\({\mathop{\rm{csch}}}\)), and the hyperbolic secant (\({\mathop{\rm{sech}}}\)).

Hyperbolic functions are related to a hyperbola in much the same way as the trigonometric functions (sometimes called circular functions) are related to the circle. Just as any point \(P\) on the unit circle \(x^{2}+y^{2}=1\) has coordinates \(( \cos t,\sin t)\), as shown in Figure 20, a point \(P\) on the hyperbola \(x^{2}-y^{2}=1\) has coordinates \((\cosh t, \sinh t)\), as shown in Figure 21. Moreover, both the sector of the circle shown in Figure 20 and the shaded portion of Figure 21 have areas that each equal \(\dfrac{t}{2}\).

The functions \(y=\sinh x\) and \(y=\cosh x\) are defined for all real numbers. The hyperbolic sine function, \(y=\sinh x\), is an odd function, so its graph is symmetric with respect to the origin; the hyperbolic cosine function, \(y=\cosh x\), is an even function, so its graph is symmetric with respect to the \(y\)-axis. Their graphs may be found by combining the graphs of \(y=\dfrac{e^{x}}{2}\) and \(y=\dfrac{e^{-x}}{2}\), as illustrated in Figures 22 and 23, respectively. The range of \(y=\sinh x\) is all real numbers, and the range of \(y=\cosh x\) is the interval \([ 1,\infty ) \).

The remaining four hyperbolic functions are combinations of the hyperbolic cosine and hyperbolic sine functions, and their relationships are similar to those of the trigonometric functions.

\[ \begin{eqnarray*} \hbox{Hyperbolic tangent:} & y&=&\tanh x=\dfrac{\sinh x}{ \cosh x}=\dfrac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\\[11pt] \hbox{Hyperbolic cotangent:} & y&=&\coth x=\dfrac{\cosh x}{\sinh x} =\dfrac{e^{x}+e^{-x}}{e^{x}-e^{-x}}\\[11pt] \hbox{Hyperbolic secant:} & y&=&\sec\!{\rm h}\,x=\dfrac{1}{ \cosh x}=\dfrac{2}{e^{x}+e^{-x}}\\[11pt] \hbox{Hyperbolic cosecant:} & y&=&\csc\!{\rm h}\,x=\dfrac{1}{\sinh x}=\dfrac{2}{e^{x}-e^{-x}} \end{eqnarray*} \]

The hyperbolic cosine function has an interesting physical interpretation. If a cable or chain of uniform density is suspended at its ends, such as with high-voltage lines, it will assume the shape of the graph of a hyperbolic cosine.

Suppose we fix our coordinate system, as in Figure 28, so that the cable, which is suspended from endpoints of equal height, lies in the \(xy\)-plane, with the lowest point of the cable at the point \(( 0,b) \). Then the shape of the cable will be modeled by the equation \[ y=a\cosh \dfrac{x}{a}+b-a \]

where \(a\) is a constant that depends on the weight per unit length of the cable and on the tension or horizontal force holding the ends of the cable. The graph of this equation is called a catenary, from the Latin word catena that means “chain.”

2 Establish Identities for Hyperbolic Functions

There are identities for the hyperbolic functions that remind us of the trigonometric identities. For example, \(\tanh x=\dfrac{\sinh x}{\cosh x}\) and \(\coth x=\dfrac{1}{\tanh x}\). Other useful hyperbolic identities include \[\bbox[5px, border:1px solid black, #F9F7ED] {\cosh ^{2}x-\sinh ^{2}x=1\qquad \tanh^{2}x+\hbox{sech}^{2}x=1\qquad \coth^{2}x-\hbox{csch}^{2}x=1} \]

Numerous other identities involving hyperbolic functions can be established. We list some below.

The derivations of these identities are left as exercises. (See Problems 17–20.)

3 Differentiate Hyperbolic Functions

Since the hyperbolic functions are algebraic combinations of \(e^{x}\) and \(e^{-x}\), they are differentiable at all real numbers for which they are defined. For example, \[ \begin{eqnarray*} \dfrac{d}{dx}\sinh x &=&\dfrac{d}{dx}\left( \dfrac{e^{x}-e^{-x}}{2}\right) = \dfrac{1}{2}\left[ \dfrac{d}{dx}e^{x}-\dfrac{d}{dx}e^{-x}\right] =\dfrac{1}{2 }( e^{x}+e^{-x}) =\cosh x\\ \dfrac{d}{dx}\cosh x&=&\dfrac{d}{dx}\left( \dfrac{e^{x}+e^{-x}}{2}\right) = \dfrac{1}{2}\left[ \dfrac{d}{dx}e^{x}+\dfrac{d}{dx}e^{-x}\right] =\dfrac{1}{2 }( e^{x}-e^{-x}) =\sinh x\\ \dfrac{d}{dx}{\mathop{\rm{csch}}}~x&=&\dfrac{d}{dx}\left( \dfrac{1}{\sinh x}\right) = \dfrac{-\dfrac{d}{dx}\sinh x}{\sinh ^{2}x}=\dfrac{-\cosh x}{\sinh ^{2}x}=- \dfrac{1}{\sinh x}\cdot \dfrac{\cosh x}{\sinh x}\\ &=&-{\mathop{\rm{csch}}}~x\coth x \end{eqnarray*} \]

The formulas for the derivatives of the hyperbolic functions are listed below. \[\bbox[5px, border:1px solid black, #F9F7ED]{ {\dfrac{d}{dx}\sinh x=\cosh x\quad\dfrac{d}{dx}\tanh x=\sec\!{\rm h}^{2}\,x\quad\dfrac{d}{dx}\csc\!{\rm h}\,x=-\csc\!{\rm h}\,x\coth x\\ \dfrac{d}{dx}\cosh x =\sinh x\quad\dfrac{d}{dx}\coth x=-\csc\!{\rm h}^{2}\,x\quad\dfrac{d}{dx}\sec\!{\rm h}\,x=-\sec\!{\rm h}\,x\tanh x} } \]

4 Differentiate Inverse Hyperbolic Functions

The graph of \(y=\sinh x\), shown in Figure 22, suggests that every horizontal line intersects the graph at exactly one point. In fact, the function \(y=\) \(\sinh x\) is one-to-one, and so it has an inverse function. We denote the inverse by \(y=\sinh ^{-1}x\) and define it as \[\bbox[5px, border:1px solid black, #F9F7ED]{ {y=\sinh ^{-1}x\quad\hbox{ if and only if }\quad x=\sinh y}} \]

The domain of \(y=\sinh ^{-1}x\) is the set of real numbers, and the range is also the set of real numbers. See Figure 30(a).

The graph of \(y=\cosh x\) (see Figure 23) shows that every horizontal line above \(y=1\) intersects the graph of \(y=\cosh x\) at two points so \(y=\cosh x\) is not one-to-one. However, if the domain of \(y=\cosh x \) is restricted to the nonnegative values of \(x\), we have a one-to-one function that has an inverse. We denote the inverse function by \(y=\) \(\cosh ^{-1}x\) and define it as \[\bbox[5px, border:1px solid black, #F9F7ED]{ {y=\cosh ^{-1}x\quad\hbox{ if and only if }\quad x=\cosh y \quad y ≥ 0}} \]

The domain of \(y=\cosh ^{-1}x\) is \(x ≥ 1\), and the range is \(y ≥ 0\). See Figure 30(b).

The other inverse hyperbolic functions are defined similarly. Their graphs are given in Figure 30(c)–(f).

Since the hyperbolic functions are defined in terms of the exponential function, the inverse hyperbolic functions can be expressed in terms of natural logarithms.

\[\bbox[5px, border:1px solid black, #F9F7ED]{ {y=\sinh ^{-1}x=\ln \big(x+\ \sqrt{x^{2}+1}\big)\qquad \hbox{for all real }x}} \] \[\bbox[5px, border:1px solid black, #F9F7ED]{ {y=\cosh ^{-1}x=\ln \big( x+\ \sqrt{x^{2}-1}\big)\qquad x ≥ 1}} \] \[\bbox[5px, border:1px solid black, #F9F7ED]{ {y=\tanh ^{-1}x=\dfrac{1}{2}\ln \left( \dfrac{1+x}{1-x}\right)\qquad \vert x\vert <1}} \] \[\bbox[5px, border:1px solid black, #F9F7ED]{ {y=\coth ^{-1}x=\dfrac{1}{2}\ln \left( \dfrac{x+1}{x-1}\right)\qquad \vert x\vert >1}} \]

We show that \(\sinh ^{-1}x=\ln \big( x+\ \sqrt{x^{2}+1}\big) \) here, and leave the others as exercises. (See Problems 61–63.)

Similarly, we have the following derivative formulas. If \(y=\cosh ^{-1}x\), then \[\bbox[5px, border:1px solid black, #F9F7ED]{ {y^\prime =\dfrac{d}{dx}\cosh ^{-1}x=\dfrac{1}{\ \sqrt{x^{2}-1}}\qquad x>1}} \]

If \(y=\tanh ^{-1}x\), then \[\bbox[5px, border:1px solid black, #F9F7ED]{ {y^\prime =\dfrac{d}{dx}\tanh ^{-1}x=\dfrac{1}{1-x^{2}}\qquad \vert x\vert <1}} \]