Higher Engineering Mathematics, Sixth Edition

Hyperbolic functions 47

Now try the following exercise

Exercise 21 Further problemson

hyperbolic identities

In Problems 1 to 4, prove the given identities.

1. (a) ch(P−Q)≡chPchQ−shPshQ
   (b) ch2x≡ch^2 x+sh^2 x


2. (a) cothx≡2cosech2x+thx
   (b) ch2θ− 1 ≡2sh^2 θ


3. (a) th(A−B)≡

thA−thB

1 −thAthB

(b) sh2A≡2shAchA

4. (a) sh(A+B)≡shAchB+chAshB

(b)

sh^2 x+ch^2 x− 1

2ch^2 xcoth^2 x

≡tanh^4 x

5. Given Pex−Qe−x≡6chx−2shx,findP
   andQ [P= 2 ,Q=−4]


6. If 5ex−4e−x≡Ashx+Bchx,findAandB.
   [A= 9 ,B=1]

5.4 Solving equations involving

hyperbolic functions

Equations such as sinhx= 3 .25 or cothx= 3 .478 may
be determined using a calculator. This is demonstrated
in Worked Problems 15 to 21.

Problem 15. Solve the equation shx=3, correct

to 4 significant figures.

If sinhx=3, thenx=sinh−^13
This can be determined by calculator.

(i) Press hyp

(ii) Choose 4, which is sinh−^1

(iii) Type in 3

(iv) Close bracket )

(v) Press=and the answer is 1.818448459

i.e. the solution of shx=3is:x= 1. 818 , correct to 4
significant figures.

Problem 16. Solve the equation chx= 1 .52,

correct to 3 decimal places.

Using acalculator with asimilar procedureas in Worked

Problem 15, check that:

x= 0. 980 ,correct to 3 decimal places.

With reference to Fig. 5.2, it can be seen that there

will be two values corresponding to y=coshx=

1 .52. Hence,x=± 0. 980

Problem 17. Solve the equation tanhθ= 0 .256,

correct to 4 significant figures.

Using acalculator with asimilar procedureas in Worked

Problem 15, check that gives

θ= 0. 2618 ,correct to 4 significant figures.

Problem 18. Solve the equation sechx= 0 .4562,

correct to 3 decimal places.

If sechx= 0 .4562, then x=sech−^10. 4562 =

cosh−^1

(

1

0. 4562

)

since cosh=

1

sech

i.e. x= 1 .421, correct to 3 decimal places.

With reference to the graph ofy=sechxin Fig. 5.4, it

can be seen that there will be two values corresponding

toy=sechx= 0. 4562

Hence,x=± 1. 421

Problem 19. Solve the equation

cosechy=− 0 .4458, correct to 4 significant figures.

If cosechy=− 0 .4458, theny=cosech−^1 (− 0. 4458 )

=sinh−^1

(

1

− 0. 4458

)

since sinh=

1

cosech

i.e. y=− 1. 547 , correct to 4 significant figures.

Problem 20. Solve the equation cothA= 2 .431,

correct to 3 decimal places.

If cothA= 2 .431, then A=coth−^12. 431 =

tanh−^1

(

1

2. 431

)

since tanh=

1

coth

i.e. A= 0. 437 , correct to 3 decimal places.

Problem 21. A chain hangs in the form given by

y=40 ch

x

40

. Determine, correct to 4 significant
figures, (a) the value ofywhenxis 25, and (b) the
value ofxwheny= 54. 30