James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 3.11 Hyperbolic Functions 259
Certain even and odd combinations of the exponential functions e x and e 2x 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. For this
reason they are collectively called hyperbolic functions and individually called hyperbolic
sine, hyperbolic cosine, and so on.
Definition of the Hyperbolic Functions
sinh x − e x 2 e 2x
2
cosh x − e x 1 e 2x
2
tanh x − sinh x
cosh x
csch x − 1
sinh x
sech x − 1
cosh x
coth x − cosh x
sinh x
The graphs of hyperbolic sine and cosine can be sketched using graphical addition as
in Figures 1 and 2.
y
y
1
y= ´
2
y=sinh x
y=cosh x
y
y=1
0
1
y=_ e–®
2
x
1
y= e–®
2
1
1
y= ´
2
0 x
y=_1
0
x
FIGURE 1
FIGURE 2
FIGURE 3
y − sinh x − 1 2 e x 2 1 2 e2x y − cosh x − 1 2 e x 1 1 2 e2x y − tanh x
y
0 x
FIGURE 4
A catenary y − c 1 a coshsxyad
Note that sinh has domain R and range R, while cosh has domain R and range f1, `d.
The graph of tanh is shown in Figure 3. It has the horizontal asymptotes y − 61. (See
Exercise 23.)
Some of the mathematical uses of hyperbolic functions will be seen in Chapter 7.
Applications to science and engineering occur whenever an entity such as light, velocity,
electricity, or radioactivity is gradually absorbed or extinguished, for the decay can be
represented by hyperbolic functions. The most famous application is the use of hyperbolic
cosine to describe the shape of a hanging wire. It can be proved that if a heavy
flexible cable (such as a telephone or power line) is suspended between two points at the
same height, then it takes the shape of a curve with equation y − c 1 a coshsxyad called
a catenary (see Figure 4). (The Latin word catena means “chain.”)
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