Higher Engineering Mathematics, Sixth Edition

Hyperbolic functions 43

Now try the following exercise

Exercise 20 Further problemson

evaluating hyperbolic functions

In Problems 1 to 6, evaluate correct to 4 significant

figures.

1. (a) sh0.64 (b) sh2. 182
   [(a) 0.6846 (b) 4.376]


2. (a) ch0.72 (b) ch2. 4625
   [(a) 1.271 (b) 5.910]


3. (a) th0.65 (b) th1. 81
   [(a) 0.5717 (b) 0.9478]


4. (a) cosech0.543 (b) cosech3. 12
   [(a) 1.754 (b) 0.08849]


5. (a) sech 0.39 (b) sech2. 367
   [(a) 0.9285 (b) 0.1859]


6. (a) coth0.444 (b) coth1. 843
   [(a) 2.398 (b) 1.051]


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

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).Findthe
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 .0andL=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).
A graph ofy=coshxmay be plottedusing calculator
values of hyperbolic functions. The curve is shown in
Fig.5.2.Sincethegraphissymmetrical about they-axis,

x

y

y 5 sinh x

10

8

6

4

2

2322

22

24

26

28

210

21 0123

Figure 5.1

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 includetransmission lines,atelegraph wireor

a fisherman’s line, and is used in the design of roofs and

arches. Graphs ofy=tanhx,y=cosechx,y=sechx

andy=cothxare deduced in Problems 7 and 8.

x

y

y 5 cosh x

10

8

6

4

2

2322210 123

Figure 5.2

Problem 7. Sketch graphs of (a)y=tanhx

and (b)y=cothxfor values ofxbetween

−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