**6.7** Hyperbolic Functions

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### Hyperbolic Functions

Certain even and odd combinations of the exponential

*functions e*^{x}*and e** ^{–x }*arise so frequently in mathematics and
its applications that they deserve to be given special names.

In many ways they are analogous to the trigonometric functions, and they have the same relationship to the

hyperbola that the trigonometric functions have to the circle.

### Hyperbolic Functions

**For this reason they are collectively called hyperbolic **
**functions and individually called hyperbolic sine, **

**hyperbolic cosine, and so on.**

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### Hyperbolic Functions

The hyperbolic functions satisfy a number of identities that are similar to well-known trigonometric identities.

We list some of them here and leave most of the proofs to the exercises.

### Example 1

Prove (a) cosh^{2}*x – sinh*^{2}*x = 1 and*
(b) 1 – tanh^{2}*x = sech*^{2}*x.*

Solution:

(a) cosh^{2}*x – sinh*^{2}*x =*

=

=

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*Example 1 – Solution*

(b) We start with the identity proved in part (a):

cosh^{2}*x – sinh*^{2}*x = 1*

If we divide both sides by cosh^{2}*x, we get*

or

cont’d

### Hyperbolic Functions

The derivatives of the hyperbolic functions are easily computed. For example,

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### Hyperbolic Functions

We list the differentiation formulas for the hyperbolic functions as Table 1.

### Example 2

Any of these differentiation rules can be combined with the Chain Rule. For instance,

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### Inverse Hyperbolic Functions

### Inverse Hyperbolic Functions

The sinh and tanh are one-to-one functions and so they
have inverse functions denoted by sinh^{–1 }and tanh^{–1}. The
cosh is not one-to-one, but when restricted to the domain
[0, ) it becomes one-to-one.

The inverse hyperbolic cosine function is defined as the inverse of this restricted function.

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### Inverse Hyperbolic Functions

We can sketch the graphs of sinh^{–1}, cosh^{–1}, and tanh^{–1 }in
Figures 8, 9, and 10.

**Figure 8**

domain = range =

**Figure 9**

domain = [1, ) range = [0, )

### Inverse Hyperbolic Functions

domain = (–1, 1) range =

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### Inverse Hyperbolic Functions

Since the hyperbolic functions are defined in terms of exponential functions, it’s not surprising to learn that the inverse hyperbolic functions can be expressed in terms of logarithms.

In particular, we have:

### Example 3

Show that sinh^{–1}*x =*
Solution:

*Let y = sinh*^{–1}*x. Then *

so *e*^{y }*– 2x – e** ^{–y}* = 0

*or, multiplying by e*

*,*

^{y}16

*Example 3 – Solution*

*This is really a quadratic equation in e** ^{y}*:

*(e*

*)*

^{y}^{2}

*– 2x(e*

*) – 1 = 0 Solving by the quadratic formula, we get*

^{y}*Note that e** ^{y}* > 0, but

cont’d

*Example 3 – Solution*

Thus the minus sign is inadmissible and we have

Therefore

This shows that

cont’d

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### Inverse Hyperbolic Functions

The inverse hyperbolic functions are all differentiable because the hyperbolic functions are differentiable.

### Example 4

Prove that Solution:

*Let y = sinh*^{–1}*x. Then sinh y = x. If we differentiate this *
*equation implicitly with respect to x, we get*

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*Example 4 – Solution*

Since cosh^{2}*y – sinh*^{2}*y = 1 and cosh y* ≥ 0, we have
*cosh y = so*

cont’d