44 Higher Engineering Mathematics
x
0
1
2
3
sh x
0 1.18 3.63 10.02
ch x
1 1.54 3.76 10.07
y = th x =
sh x
ch x
0 0.77 0.97 0.995
y = coth x =
ch x
sh x
±∞ 1.31 1.04 1.005
(a) A graph of y = tanh x is shown in Fig. 5.3(a)
(b) A graph of y = coth x is shown in Fig. 5.3(b)
Both graphs are symmetrical about the origin thus tanh x
and coth x are odd functions.
Problem 8. Sketch graphs of (a) y = cosech x
and (b) y = sech x from x =−4 to x = 4, and, from
the graphs, determine whether they are odd or
even functions.
y
x
(a)
y 5 tanh x
23 22 21
1
21
1 2 3
0
y
x
(b)
y 5coth x
y 5 coth x
23 22 21
1 2 3
0
1
2
3
21
23
22
Figure 5.3
A table of values is drawn up as shown below
x
−4
−3
−2
−1
sh x
−22.29 −10.02 −3.63 −1.18
cosech x =
1
sh x
−0.04 −0.10 −0.28 −0.85
ch x
27.31
10.07
3.76
1.54
sech x =
1
ch x
0.04
0.10
0.27
0.65
x
0
1
2
3
4
sh x
0
1.18 3.63 10.02 27.29
cosech x =
1
sh x
±∞ 0.85 0.28 0.10 0.04
ch x
1
1.54 3.76 10.07 27.31
sech x =
1
ch x
1
0.65 0.27 0.10 0.04
(a) A graph of y = cosech x is shown in Fig. 5.4(a).
The graph is symmetrical about the origin and is
thus an odd function.
(b) A graph of y = sech x is shown in Fig. 5.4(b). The
graph is symmetrical about the y-axis and is thus
an even function.
y 5 cosech x
y
x
(a)
y 5 cosech x
232221
1 2 3
0
1
2
3
21
23
22
y
x
(b)
y 5 sech x
232221
1 2 3
0
1
Figure 5.4