44 Higher Engineering Mathematics
x 0 1 2 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=tanhxisshowninFig.5.3(a)
(b) A graph ofy=cothxisshowninFig.5.3(b)
Both graphs are symmetrical about the originthus tanhx
and cothxare odd functions.
Problem 8. Sketch graphs of (a)y=cosechx
and (b)y=sechxfromx=−4tox=4, and, from
the graphs, determine whether they are odd or
even functions.
y
x
(a)
y 5 tanh x
232221
1
21
0123
y
x
(b)
y 5 coth x
y 5 coth x
2322211230
1
2
3
21
23
22
Figure 5.3
A table of values is drawn up as shown below
x − 4 − 3 − 2 − 1
shx −22.29 −10.02 −3.63 −1.18
cosechx=
1
shx
−0.04 −0.10 −0.28 −0.85
chx 27.31 10.07 3.76 1.54
sechx=
1
chx
0.04 0.10 0.27 0.65
x 0 1 2 3 4
shx 0 1.18 3.63 10.02 27.29
cosechx=
1
shx
±∞ 0.85 0.28 0.10 0.04
chx 1 1.54 3.76 10.07 27.31
sechx=
1
chx
1 0.65 0.27 0.10 0.04
(a) A graph ofy=cosechxis shown in Fig. 5.4(a).
The graph is symmetrical about the origin and is
thus anodd function.
(b) A graph ofy=sechxisshowninFig.5.4(b).The
graph is symmetrical about they-axis and is thus
aneven function.
y 5 cosech x
y
x
(a)
y 5 cosech x
232221 0123
1
2
3
21
23
22
y
x
(b)
y 5 sech x
2322210 123
1
Figure 5.4