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Mathematics LibreTexts

4.11: Hyperbolic Functions

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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

coshx=ex+ex2,

and the hyperbolic sine is the function

sinhx=exex2.

Notice that cosh is even (that is, cosh(x)=cosh(x)) while sinh is odd (sinh(x)=sinh(x)), and coshx+sinhx=ex. Also, for all x, coshx>0, while sinhx=0 if and only if exex=0, which is true precisely when x=0.

Lemma 4.11.2

The range of coshx is [1,).

Proof

Let y=coshx. We solve for x:

y=ex+ex22y=ex+ex2yex=e2x+10=e2x2yex+1ex=2y±4y242ex=y±y21

From the last equation, we see y21, and since y0, it follows that y1.

Now suppose y1, so y±y21>0. Then x=ln(y±y21) is a real number, and y=coshx, so y is in the range of cosh(x).

Definition 4.11.3: Hyperbolic Tangent and Cotangent

The other hyperbolic functions are

tanhx=sinhxcoshxcothx=coshxsinhxsechx=1coshxcschx=1sinhx

The domain of coth and csch is x0 while the domain of the other hyperbolic functions is all real numbers. Graphs are shown in Figure 4.11.1.

alt

Figure 4.11.1: The hyperbolic functions: cosh, sinh, tanh, sech, csch, coth.

Certainly the hyperbolic functions do not closely resemble the trigonometric functions graphically. But they do have analogous properties, beginning with the following identity.

Theorem 4.11.4

For all x in R, cosh2xsinh2x=1.

Proof

The proof is a straightforward computation:

cosh2xsinh2x=(ex+ex)24(exex)24=e2x+2+e2xe2x+2e2x4=44=1.

This immediately gives two additional identities:

1tanh2x=sech2xandcoth2x1=csch2x.

The identity of the theorem also helps to provide a geometric motivation. Recall that the graph of x2y2=1 is a hyperbola with asymptotes x=±y whose x-intercepts are ±1. If (x,y) is a point on the right half of the hyperbola, and if we let x=cosht, then y=±x21=±cosh2x1=±sinht. So for some suitable t, cosht and sinht are the coordinates of a typical point on the hyperbola. In fact, it turns out that t is twice the area shown in the first graph of Figure 4.11.2. Even this is analogous to trigonometry; cost and sint are the coordinates of a typical point on the unit circle, and t is twice the area shown in the second graph of Figure 4.11.2.

alt

Figure 4.11.2: Geometric definitions of cosh, sinh, cos, sin: t is twice the shaded area in each figure.

Given the definitions of the hyperbolic functions, finding their derivatives is straightforward. Here again we see similarities to the trigonometric functions.

Theorem 4.11.5

ddxcoshx=sinhx and ddxsinhx=coshx.

Proof

ddxcoshx=ddxex+ex2=exex2=sinhx,

and

ddxsinhx=ddxexex2=ex+ex2=coshx.

Since coshx>0, sinhx is increasing and hence injective, so sinhx has an inverse, arcsinhx. Also, sinhx>0 when x>0, so coshx is injective on [0,) and has a (partial) inverse, arccoshx. The other hyperbolic functions have inverses as well, though arcsechx is only a partial inverse. We may compute the derivatives of these functions as we have other inverse functions.

Theorem 4.11.6

ddxarcsinhx=11+x2.

Proof

Let y=arcsinhx, so sinhy=x. Then

ddxsinhy=cosh(y)y=1,

and so

y=1coshy=11+sinh2y=11+x2.

The other derivatives are left to the exercises.

Exercises 4.11.

Exercise 4.11.1

Show that the range of sinhx is all real numbers. (Hint: show that if y=sinhx, then x=ln(y+y2+1).)

Exercise 4.11.2

Ex 4.11.2 Compute the following limits: limxcoshx,limxsinhx,limxtanhx,limx(coshxsinhx).

Answer

,,1,0.

Exercise 4.11.3

Show that the range of tanhx is (1,1). What are the ranges of coth,sech, and csch? (Use the fact that they are reciprocal functions.)

Answer

Range of y=coth(x) is (,1)(1,), that is, y[1,1].

Range of y=sech(x) is (0,1],that is, 0<y1.

Range of y=csch(x) is y0.

Exercise 4.11.4

Prove that for every pair of reals x,yR, sinh(x+y)=sinhxcoshy+coshxsinhy. Obtain a similar identity for sinh(xy).

Answer

sinh(xy)=sinhxcoshycoshxsinhy

Exercise 4.11.5

Prove that for every pair of reals x,yR, cosh(x+y)=coshxcoshy+sinhxsinhy. Obtain a similar identity for cosh(xy).

Answer

cosh(xy)=coshxcoshysinhxsinhy

Exercise 4.11.6

Use exercises 4 and 5 to show that sinh(2x)=2sinhxcoshx and cosh(2x)=cosh2x+sinh2x for every x.

Conclude also that (cosh(2x)1)/2=sinh2x.

Exercise 4.11.7

Show that ddx(tanhx)=sech2x.

Compute the derivatives of the remaining hyperbolic functions as well.

Answer

ddx(sechx)=sechxtanhx;ddx(cschx)=cschxcothx;ddx(cothx)=csch2x.

Exercise 4.11.8

What are the domains of the six inverse hyperbolic functions?

Answer

Domain of arccoshx=[1,)

Domain of arcsinhx=R=(,)

Domain of arctanhx=(1,1)

Domain of arcsechx=(0,1]

Domain of arccschx=x0

Domain of arccothx=(,1)(1,), that is, x[1,1].

Exercise 4.11.9

Sketch the graphs of all six inverse hyperbolic functions.

Contributors


This page titled 4.11: Hyperbolic Functions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Terry Betteridge, Editor via source content that was edited to the style and standards of the LibreTexts platform.

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