Higher Engineering Mathematics, Sixth Edition

160 Higher Engineering Mathematics

But from Chapter 5, Problem 6,

cosh^2 θ−sinh^2 θ= 1 ,

hence cos^2 jθ+sin^2 jθ=^1

Problem 2. Verify that sinj 2 A=2sinjAcosjA.

Fromequation(6),writing2Aforθ,sinj 2 A=jsinh2A,

and from Chapter 5, Table 5.1, page 45, sinh2A=

2sinhAcoshA.

Hence, sinj^2 A= j(2sinhAcoshA)

But, sinhA=^12 (eA−e−A)and coshA=^12 (eA+e−A)

Hence, sinj 2 A=j 2

(

eA−e−A

2

)(

eA+e−A

2

)

=−

2

j

(

eA−e−A

2

)(

eA+e−A

2

)

=−

2

j

(

sinjθ

j

)

(cosjθ)

=2sinjAcosjAsince j^2 =− 1

i.e. sinj 2 A=2sinjAcosjA

Now try the following exercise

Exercise 70 Further problems on the

relationship between trigonometric and

hyperbolic functions

Verify the following identities by expressing in

exponential form.

1. sinj(A+B)=sinjAcosjB+cosjAsinjB


2. cosj(A−B)=cosjAcosjB+sinjAsinjB


3. cosj 2 A= 1 −2sin^2 jA


4. sinjAcosjB=^12 [sinj(A+B)+sinj(A−B)]


5. sinjA−sinjB

=2cosj

(

A+B

2

)

sinj

(

A−B

2

)

16.2 Hyperbolic identities

From Chapter 5, coshθ=^12 (eθ+e−θ)

Substitutingjθforθgives:

coshjθ=^12 (ejθ+e−jθ)=cosθ,from equation (3),

i.e. coshjθ=cosθ (7)

Similarly, from Chapter 5,

sinhθ=^12 (eθ−e−θ)

Substitutingjθforθgives:

sinhjθ=^12 (ejθ−e−jθ)=jsinθ, from equation (4).

Hence sinhjθ=jsinθ (8)

tanjθ=

sinjθ

coshjθ

From equations (5) and (6),

sinjθ

cosjθ

=

jsinhθ

coshθ

=jtanhθ

Hence tanjθ=jtanhθ (9)

Similarly, tanhjθ=

sinhjθ

coshjθ

From equations (7) and (8),

sinhjθ

coshjθ

=

jsinθ

cosθ

=jtanθ

Hence tanhjθ=jtanθ (10)

Two methods are commonly used to verify hyperbolic

identities. These are (a) by substitutingjθ(andjφ)in

the corresponding trigonometric identity and using the

relationships given in equations (5) to (10) (see Prob-

lems 3 to 5) and (b) by applying Osborne’s rule given

in Chapter 5, page 45.

Problem 3. By writing jAforθin cot^2 θ+ 1 =

cosec^2 θ, determine the corresponding hyperbolic

identity.

Substituting jAforθgives:

cot^2 jA+ 1 =cosec^2 jA,

i.e. cos

(^2) jA
sin^2 jA

• 1 =
  1
  sin^2 jA