Higher Engineering Mathematics

HYPERBOLIC FUNCTIONS 43

A

7. A telegraph wire hangs so that its shape is
   described byy=50 ch

x

50

. Evaluate, correct
to 4 significant figures, the value ofywhen
x=25. [56.38]
8. The lengthlof a heavy cable hanging under
gravity is given byl= 2 csh (L/ 2 c). Find the
value oflwhenc=40 andL=30.
[30.71]

9.V^2 = 0. 55 Ltanh (6.3d/L) is a formula for

velocityVof waves over the bottom of shal-

low water, wheredis the depth andLis the

wavelength. If d= 8 .0 andL=96, calculate

the value ofV. [5.042]

5.2 Graphs of hyperbolic functions

A graph ofy=sinhxmay be plotted using calculator
values of hyperbolic functions. The curve is shown
in Fig. 5.1. Since the graph is symmetrical about
the origin, sinhxis anodd function(as stated in
Section 5.1).

− 3 −^2 −^1 01 2 3

− 2

− 4

− 6

− 8

2

4

6

8

10

− 10

x

y

y = sinh x

Figure 5.1

A graph ofy=coshxmay be plotted using cal-
culator values of hyperbolic functions. The curve is
shown in Fig. 5.2. Since the graph is symmetrical
about they-axis, coshxis aneven function(as stated
in Section 5.1). The shape ofy=coshxis that of a
heavy rope or chain hanging freely under gravity and
is called acatenary. Examples include transmission
lines, a telegraph wire or a fisherman’s line, and is
used in the design of roofs and arches. Graphs of
y=tanhx,y=cosechx,y=sechxandy=cothx
are deduced in Problems 4 and 5.

− 3 − 2 − 10 123

2

4

6

8

10

x

y

y = cosh x

Figure 5.2

Problem 4. Sketch graphs of (a) y=tanhx

and (b) y=cothx for values of x between

−3 and 3.

A table of values is drawn up as shown below

x − 3 − 2 − 1

shx −10.02 −3.63 −1.18

chx 10.07 3.76 1.54

y=thx=

shx

chx

−0.995 −0.97 −0.77

y=cothx=

chx

shx

−1.005 −1.04 −1.31

x 012 3

shx 0 1.18 3.63 10.02

chx 1 1.54 3.76 10.07

y=thx=

shx

chx

0 0.77 0.97 0.995

y=cothx=

chx

shx

±∞ 1.31 1.04 1.005

(a) A graph ofy=tanhxis shown in Fig. 5.3(a)

(b) A graph ofy=cothxis shown in Fig. 5.3(b)

Both graphs are symmetrical about the origin thus

tanhxand cothxare odd functions.

Problem 5. Sketch graphs of (a)y=cosechx

and (b)y=sechxfromx=−4tox=4, and,

from the graphs, determine whether they are odd

or even functions.