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# 4.11: Hyperbolic Functions

The hyperbolic functions appear with some frequency in applications, and are quite similar in many respects to the trigonometric functions. This is a bit surprising given our initial definitions.

Definition 4.11.1: Hyperbolic Cosines and Sines

The hyperbolic cosine is the function

$\cosh x ={e^x +e^{-x }\over2},$

and the hyperbolic sine is the function

$\sinh x ={e^x -e^{-x}\over 2}.$

Notice that $$\cosh$$ is even (that is, $$\cosh(-x)=\cosh(x)$$) while $$\sinh$$ is odd ($$\sinh(-x)=-\sinh(x)$$), and $$\cosh x + \sinh x = e^x$$. Also, for all $$x$$, $$\cosh x >0$$, while $$\sinh x=0$$ if and only if $$e^x -e^{-x }=0$$, which is true precisely when $$x=0$$.

Lemma 4.11.2

The range of $$\cosh x$$ is $$[1,\infty)$$.

Proof

Let $$y= \cosh x$$. We solve for $$x$$:

\eqalign{y&={e^x +e^{-x }\over 2}\cr 2y &= e^x + e^{-x }\cr 2ye^x &= e^{2x} + 1\cr 0 &= e^{2x}-2ye^x +1\cr e^{x} &= {2y \pm \sqrt{4y^2 -4}\over 2}\cr e^{x} &= y\pm \sqrt{y^2 -1}\cr}

From the last equation, we see $$y^2 \geq 1$$, and since $$y\geq 0$$, it follows that $$y\geq 1$$.

Now suppose $$y\geq 1$$, so $$y\pm \sqrt{y^2 -1}>0$$. Then $$x = \ln(y\pm \sqrt{y^2 -1})$$ is a real number, and $$y =\cosh x$$, so $$y$$ is in the range of $$\cosh(x)$$.

$$\square$$

Definition 4.11.3: Hyperbolic Tangent and Cotangent

The other hyperbolic functions are

\eqalign{\tanh x &= {\sinh x\over\cosh x}\cr \coth x &= {\cosh x\over\sinh x}\cr \text{sech} x &= {1\over\cosh x}\cr \text{csch} x &= {1\over\sinh x}\cr}

The domain of $$\coth$$ and $$\text{csch}$$ is $$x\neq 0$$ while the domain of the other hyperbolic functions is all real numbers. Graphs are shown in Figure $$\PageIndex{1}$$