Higher Engineering Mathematics, Sixth Edition

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