# 1.5 Exponential and logarithmic functions  (Page 6/17)

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Compare the relative severity of a magnitude $8.4$ earthquake with a magnitude $7.4$ earthquake.

The magnitude $8.4$ earthquake is roughly $10$ times as severe as the magnitude $7.4$ earthquake.

## Hyperbolic functions

The hyperbolic functions are defined in terms of certain combinations of ${e}^{x}$ and ${e}^{\text{−}x}.$ These functions arise naturally in various engineering and physics applications, including the study of water waves and vibrations of elastic membranes. Another common use for a hyperbolic function is the representation of a hanging chain or cable, also known as a catenary ( [link] ). If we introduce a coordinate system so that the low point of the chain lies along the $y$ -axis, we can describe the height of the chain in terms of a hyperbolic function. First, we define the hyperbolic functions    .

## Definition

Hyperbolic cosine

$\text{cosh}\phantom{\rule{0.1em}{0ex}}x=\frac{{e}^{x}+{e}^{\text{−}x}}{2}$

Hyperbolic sine

$\text{sinh}\phantom{\rule{0.1em}{0ex}}x=\frac{{e}^{x}-{e}^{\text{−}x}}{2}$

Hyperbolic tangent

$\text{tanh}\phantom{\rule{0.1em}{0ex}}x=\frac{\text{sinh}\phantom{\rule{0.1em}{0ex}}x}{\text{cosh}\phantom{\rule{0.1em}{0ex}}x}=\frac{{e}^{x}-{e}^{\text{−}x}}{{e}^{x}+{e}^{\text{−}x}}$

Hyperbolic cosecant

$\text{csch}\phantom{\rule{0.1em}{0ex}}x=\frac{1}{\text{sinh}\phantom{\rule{0.1em}{0ex}}x}=\frac{2}{{e}^{x}-{e}^{\text{−}x}}$

Hyperbolic secant

$\text{sech}\phantom{\rule{0.1em}{0ex}}x=\frac{1}{\text{cosh}\phantom{\rule{0.1em}{0ex}}x}=\frac{2}{{e}^{x}+{e}^{\text{−}x}}$

Hyperbolic cotangent

$\text{coth}\phantom{\rule{0.1em}{0ex}}x=\frac{\text{cosh}\phantom{\rule{0.1em}{0ex}}x}{\text{sinh}\phantom{\rule{0.1em}{0ex}}x}=\frac{{e}^{x}+{e}^{\text{−}x}}{{e}^{x}-{e}^{\text{−}x}}$

The name cosh rhymes with “gosh,” whereas the name sinh is pronounced “cinch.” Tanh , sech , csch , and coth are pronounced “tanch,” “seech,” “coseech,” and “cotanch,” respectively.

Using the definition of $\text{cosh}\left(x\right)$ and principles of physics, it can be shown that the height of a hanging chain, such as the one in [link] , can be described by the function $h\left(x\right)=a\phantom{\rule{0.1em}{0ex}}\text{cosh}\left(x\text{/}a\right)+c$ for certain constants $a$ and $c.$

But why are these functions called hyperbolic functions ? To answer this question, consider the quantity ${\text{cosh}}^{2}t-{\text{sinh}}^{2}t.$ Using the definition of $\text{cosh}$ and $\text{sinh},$ we see that

${\text{cosh}}^{2}t-{\text{sinh}}^{2}t=\frac{{e}^{2t}+2+{e}^{-2t}}{4}-\frac{{e}^{2t}-2+{e}^{-2t}}{4}=1.$

This identity is the analog of the trigonometric identity ${\text{cos}}^{2}t+{\text{sin}}^{2}t=1.$ Here, given a value $t,$ the point $\left(x,y\right)=\left(\text{cosh}\phantom{\rule{0.1em}{0ex}}t,\text{sinh}\phantom{\rule{0.1em}{0ex}}t\right)$ lies on the unit hyperbola ${x}^{2}-{y}^{2}=1$ ( [link] ).

## Graphs of hyperbolic functions

To graph $\text{cosh}\phantom{\rule{0.1em}{0ex}}x$ and $\text{sinh}\phantom{\rule{0.1em}{0ex}}x,$ we make use of the fact that both functions approach $\left(1\text{/}2\right){e}^{x}$ as $x\to \infty ,$ since ${e}^{\text{−}x}\to 0$ as $x\to \infty .$ As $x\to \text{−}\infty ,\text{cosh}\phantom{\rule{0.1em}{0ex}}x$ approaches $1\text{/}2{e}^{\text{−}x},$ whereas $\text{sinh}\phantom{\rule{0.1em}{0ex}}x$ approaches $-1\text{/}2{e}^{\text{−}x}.$ Therefore, using the graphs of $1\text{/}2{e}^{x},1\text{/}2{e}^{\text{−}x},$ and $\text{−}1\text{/}2{e}^{\text{−}x}$ as guides, we graph $\text{cosh}\phantom{\rule{0.1em}{0ex}}x$ and $\text{sinh}\phantom{\rule{0.1em}{0ex}}x.$ To graph $\text{tanh}\phantom{\rule{0.1em}{0ex}}x,$ we use the fact that $\text{tanh}\left(0\right)=1,-1<\text{tanh}\left(x\right)<1$ for all $x,\text{tanh}\phantom{\rule{0.1em}{0ex}}x\to 1$ as $x\to \infty ,$ and $\text{tanh}\phantom{\rule{0.1em}{0ex}}x\to \text{−}1$ as $x\to \text{−}\infty .$ The graphs of the other three hyperbolic functions can be sketched using the graphs of $\text{cosh}\phantom{\rule{0.1em}{0ex}}x,\text{sinh}\phantom{\rule{0.1em}{0ex}}x,$ and $\text{tanh}\phantom{\rule{0.1em}{0ex}}x$ ( [link] ).

## Identities involving hyperbolic functions

The identity ${\text{cosh}}^{2}t-{\text{sinh}}^{2}t,$ shown in [link] , is one of several identities involving the hyperbolic functions, some of which are listed next. The first four properties follow easily from the definitions of hyperbolic sine and hyperbolic cosine. Except for some differences in signs, most of these properties are analogous to identities for trigonometric functions.

## Rule: identities involving hyperbolic functions

1. $\text{cosh}\left(\text{−}x\right)=\text{cosh}\phantom{\rule{0.1em}{0ex}}x$
2. $\text{sinh}\left(\text{−}x\right)=\text{−}\text{sinh}\phantom{\rule{0.1em}{0ex}}x$
3. $\text{cosh}\phantom{\rule{0.1em}{0ex}}x+\text{sinh}\phantom{\rule{0.1em}{0ex}}x={e}^{x}$
4. $\text{cosh}\phantom{\rule{0.1em}{0ex}}x-\text{sinh}\phantom{\rule{0.1em}{0ex}}x={e}^{\text{−}x}$
5. ${\text{cosh}}^{2}x-{\text{sinh}}^{2}x=1$
6. $1-{\text{tanh}}^{2}x={\text{sech}}^{2}x$
7. ${\text{coth}}^{2}x-1={\text{csch}}^{2}x$
8. $\text{sinh}\left(x±y\right)=\text{sinh}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{cosh}\phantom{\rule{0.1em}{0ex}}y±\text{cosh}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{sinh}\phantom{\rule{0.1em}{0ex}}y$
9. $\text{cosh}\left(x±y\right)=\text{cosh}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{cosh}\phantom{\rule{0.1em}{0ex}}y±\text{sinh}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{sinh}\phantom{\rule{0.1em}{0ex}}y$

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