42 Higher Engineering Mathematics
Hence sinh x and tanh x are both odd functions
(see Section 5.1), as also are cosech x
=
1
sinh x
and
coth x
=
1
tanh x
If a function of x, f (−x) = f (x), then f (x) is
called an even function of x. Replacing x by −x in
equation (2) gives:
cosh(−x) =
e −x + e −(−x)
2
=
e −x + e x
2
= cosh x
Hence cosh x is an even function (see Section 5.2), as
also is sech x
=
1
cosh x
Hyperbolic functions may be evaluated easiest using a
calculator. Many scientific notation calculators actually
possess sinh and cosh functions; however, if a calculator
does not contain these functions, then the definitions
given above may be used.
Problem 1. Evaluate sinh 5.4, correct to 4
significant figures.
Using a calculator,
(i) press hyp
(ii) press 1 and sinh( appears
(iii) type in 5.4
(iv) press ) to close the brackets
(v) press = and 110.7009498 appears
Hence, sinh 5.4 = 110.7, correct to 4 significant figures.
Alternatively, sinh 5.4 =
1
2
(e
5.4
− e
−5.4
)
=
1
2
(221.406416 ... − 0.00451658 ...)
=
1
2
(221.401899 ...)
= 110.7, correct to 4 significant figures.
Problem 2. Evaluate cosh 1.86, correct to 3
decimal places.
Using a calculator with the procedure similar to that
used in Worked Problem 1,
cosh 1.86 = 3.290, correct to 3 decimal places.
Problem 3. Evaluate th 0.52, correct to 4
significant figures.
Using a calculator with the procedure similar to that
used in Worked Problem 1,
th 0.52 = 0.4777, correct to 4 significant figures.
Problem 4. Evaluate cosech 1.4, correct to 4
significant figures.
cosech 1.4 =
1
sinh 1.4
Using a calculator,
(i) press hyp
(ii) press 1 and sinh( appears
(iii) type in 1.4
(iv) press ) to close the brackets
(v) press = and 1.904301501 appears
(vi) press x −1
(vii) press = and 0.5251269293 appears
Hence, cosech 1.4 = 0.5251, correct to 4 significant
figures.
Problem 5. Evaluate sech 0.86, correct to 4
significant figures.
sech 0.86 =
1
cosh 0.86
Using a calculator with the procedure similar to that
used in Worked Problem 4,
sech 0.86 = 0.7178, correct to 4 significant figures.
Problem 6. Evaluate coth 0.38, correct to 3
decimal places.
coth 0.38 =
1
tanh 0.38
Using a calculator with the procedure similar to that
used in Worked Problem 4,
coth 0.38 = 2.757, correct to 3 decimal places.
Hence sinh x and tanh x are both odd functions
(see Section 5.1), as also are cosech x
=
1
sinh x
and
coth x
=
1
tanh x
If a function of x, f (−x) = f (x), then f (x) is
called an even function of x. Replacing x by −x in
equation (2) gives:
cosh(−x) =
e −x + e −(−x)
2
=
e −x + e x
2
= cosh x
Hence cosh x is an even function (see Section 5.2), as
also is sech x
=
1
cosh x
Hyperbolic functions may be evaluated easiest using a
calculator. Many scientific notation calculators actually
possess sinh and cosh functions; however, if a calculator
does not contain these functions, then the definitions
given above may be used.
Problem 1. Evaluate sinh 5.4, correct to 4
significant figures.
Using a calculator,
(i) press hyp
(ii) press 1 and sinh( appears
(iii) type in 5.4
(iv) press ) to close the brackets
(v) press = and 110.7009498 appears
Hence, sinh 5.4 = 110.7, correct to 4 significant figures.
Alternatively, sinh 5.4 =
1
2
(e
5.4
− e
−5.4
)
=
1
2
(221.406416 ... − 0.00451658 ...)
=
1
2
(221.401899 ...)
= 110.7, correct to 4 significant figures.
Problem 2. Evaluate cosh 1.86, correct to 3
decimal places.
Using a calculator with the procedure similar to that
used in Worked Problem 1,
cosh 1.86 = 3.290, correct to 3 decimal places.
Problem 3. Evaluate th 0.52, correct to 4
significant figures.
Using a calculator with the procedure similar to that
used in Worked Problem 1,
th 0.52 = 0.4777, correct to 4 significant figures.
Problem 4. Evaluate cosech 1.4, correct to 4
significant figures.
cosech 1.4 =
1
sinh 1.4
Using a calculator,
(i) press hyp
(ii) press 1 and sinh( appears
(iii) type in 1.4
(iv) press ) to close the brackets
(v) press = and 1.904301501 appears
(vi) press x −1
(vii) press = and 0.5251269293 appears
Hence, cosech 1.4 = 0.5251, correct to 4 significant
figures.
Problem 5. Evaluate sech 0.86, correct to 4
significant figures.
sech 0.86 =
1
cosh 0.86
Using a calculator with the procedure similar to that
used in Worked Problem 4,
sech 0.86 = 0.7178, correct to 4 significant figures.
Problem 6. Evaluate coth 0.38, correct to 3
decimal places.
coth 0.38 =
1
tanh 0.38
Using a calculator with the procedure similar to that
used in Worked Problem 4,
coth 0.38 = 2.757, correct to 3 decimal places.