Hyperbolic functions 43
Now try the following exercise
Exercise 20 Further problems on
evaluating hyperbolic functions
In Problems 1 to 6, evaluate correct to 4 significant
figures.
1. (a) sh 0.64 (b) sh 2.182
[(a) 0.6846 (b) 4.376]
2. (a) ch 0.72 (b) ch 2.4625
[(a) 1.271 (b) 5.910]
3. (a) th 0.65 (b) th 1.81
[(a) 0.5717 (b) 0.9478]
4. (a) cosech 0.543 (b) cosech 3.12
[(a) 1.754 (b) 0.08849]
5. (a) sech 0.39 (b) sech 2.367
[(a) 0.9285 (b) 0.1859]
6. (a) coth 0.444 (b) coth 1.843
[(a) 2.398 (b) 1.051]
7. A telegraph wire hangs so that its shape is
described by y = 50 ch
x
50
. Evaluate, correct
to 4 significant figures, the value of y when
x = 25.
[56.38]
8. The length l of a heavy cable hanging under
gravity is given by l = 2c sh (L/2c). Find the
value of l when c = 40 and L =30.
[30.71]
9. V 2 = 0.55L tanh (6.3 d/L) is a formula for
velocity V of waves over the bottom of shallow water, where d is the depth and L is the
wavelength. If d = 8.0 and L =96, calculate
the value of V .
[5.042]
5.2 Graphs of hyperbolic functions
A graph of y = sinhx may be plotted using calculator
values of hyperbolic functions. The curve is shown in
Fig. 5.1. Since the graph is symmetrical about the origin,
sinh x is an odd function (as stated in Section 5.1).
A graph of y = cosh x may be plotted using calculator
values of hyperbolic functions. The curve is shown in
Fig. 5.2. Since the graph is symmetrical about the y-axis,
x
y
y 5sinh x
10
8
6
4
2
23 22 22
24
26
28
210
1 2 3
0
21
Figure 5.1
cosh x is an even function (as stated in Section 5.1).
The shape of y = cosh x is that of a heavy rope or chain
hanging freely under gravity and is called a catenary.
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 = tanh x, y = cosech x, y = sech x
and y = coth x are deduced in Problems 7 and 8.
x
y
y 5cosh x
10
8
6
4
2
23 22 21
1 2 3
0
Figure 5.2
Problem 7. Sketch graphs of (a) y = tanh x
and (b) y = coth x for values of x between
−3 and 3.
A table of values is drawn up as shown below
x
−3
−2
−1
sh x
−10.02 −3.63 −1.18
ch x
10.07
3.76
1.54
y = th x =
sh x
ch x
−0.995 −0.97 −0.77
y = coth x =
ch x
sh x
−1.005 −1.04 −1.31
Now try the following exercise
Exercise 20 Further problems on
evaluating hyperbolic functions
In Problems 1 to 6, evaluate correct to 4 significant
figures.
1. (a) sh 0.64 (b) sh 2.182
[(a) 0.6846 (b) 4.376]
2. (a) ch 0.72 (b) ch 2.4625
[(a) 1.271 (b) 5.910]
3. (a) th 0.65 (b) th 1.81
[(a) 0.5717 (b) 0.9478]
4. (a) cosech 0.543 (b) cosech 3.12
[(a) 1.754 (b) 0.08849]
5. (a) sech 0.39 (b) sech 2.367
[(a) 0.9285 (b) 0.1859]
6. (a) coth 0.444 (b) coth 1.843
[(a) 2.398 (b) 1.051]
7. A telegraph wire hangs so that its shape is
described by y = 50 ch
x
50
. Evaluate, correct
to 4 significant figures, the value of y when
x = 25.
[56.38]
8. The length l of a heavy cable hanging under
gravity is given by l = 2c sh (L/2c). Find the
value of l when c = 40 and L =30.
[30.71]
9. V 2 = 0.55L tanh (6.3 d/L) is a formula for
velocity V of waves over the bottom of shallow water, where d is the depth and L is the
wavelength. If d = 8.0 and L =96, calculate
the value of V .
[5.042]
5.2 Graphs of hyperbolic functions
A graph of y = sinhx may be plotted using calculator
values of hyperbolic functions. The curve is shown in
Fig. 5.1. Since the graph is symmetrical about the origin,
sinh x is an odd function (as stated in Section 5.1).
A graph of y = cosh x may be plotted using calculator
values of hyperbolic functions. The curve is shown in
Fig. 5.2. Since the graph is symmetrical about the y-axis,
x
y
y 5sinh x
10
8
6
4
2
23 22 22
24
26
28
210
1 2 3
0
21
Figure 5.1
cosh x is an even function (as stated in Section 5.1).
The shape of y = cosh x is that of a heavy rope or chain
hanging freely under gravity and is called a catenary.
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 = tanh x, y = cosech x, y = sech x
and y = coth x are deduced in Problems 7 and 8.
x
y
y 5cosh x
10
8
6
4
2
23 22 21
1 2 3
0
Figure 5.2
Problem 7. Sketch graphs of (a) y = tanh x
and (b) y = coth x for values of x between
−3 and 3.
A table of values is drawn up as shown below
x
−3
−2
−1
sh x
−10.02 −3.63 −1.18
ch x
10.07
3.76
1.54
y = th x =
sh x
ch x
−0.995 −0.97 −0.77
y = coth x =
ch x
sh x
−1.005 −1.04 −1.31