Higher Engineering Mathematics, Sixth Edition

42 Higher Engineering Mathematics

Hencesinhxand tanhxare both odd functions

(see Section 5.1), as also are cosechx

(

=

1

sinhx

)

and

cothx

(

=

1

tanhx

)

If a function of x, f(−x)=f(x),thenf(x) is

called aneven functionofx. Replacingxby−xin

equation (2) gives:

cosh(−x)=

e−x+e−(−x)

2

=

e−x+ex

2

=coshx

Hencecoshxis an even function(see Section 5.2), as

also is sechx

(

=

1

coshx

)

Hyperbolicfunctionsmaybeevaluatedeasiest usinga

calculator. Many scientific notation calculators actually

possess sinh and cosh functions;however, ifa calculator

does not contain these functions, then the definitions

given above may be used.

Problem 1. Evaluate sinh5.4, 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,sinh5. 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 cosh1.86, correct to 3

decimal places.

Using a calculator with the procedure similar to that

used in Worked Problem 1,

cosh1.86=3.290,correct to 3 decimal places.

Problem 3. Evaluate th0.52, correct to 4

significant figures.

Using a calculator with the procedure similar to that

used in Worked Problem 1,

th0.52=0.4777,correct to 4 significant figures.

Problem 4. Evaluate cosech1.4, correct to 4

significant figures.

cosech1. 4 =

1

sinh1. 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) pressx−^1

(vii) press=and 0.5251269293 appears

Hence, cosech1.4=0.5251, correct to 4 significant

figures.

Problem 5. Evaluate sech0.86, correct to 4

significant figures.

sech0. 86 =

1

cosh0. 86

Using a calculator with the procedure similar to that

used in Worked Problem 4,

sech0.86=0.7178,correct to 4 significant figures.

Problem 6. Evaluate coth0.38, correct to 3

decimal places.

coth0. 38 =

1

tanh0. 38

Using a calculator with the procedure similar to that

used in Worked Problem 4,

coth0.38=2.757,correct to 3 decimal places.