Hyperbolic functions 47
Now try the following exercise
Exercise 21 Further problems on
hyperbolic identities
In Problems 1 to 4, prove the given identities.
1. (a) ch (P − Q) ≡ ch P ch Q − sh P sh Q
(b) ch 2x ≡ ch 2 x + sh 2 x
2. (a) coth x ≡ 2 cosech 2x + th x
(b) ch 2θ − 1 ≡2 sh 2 θ
3. (a) th (A − B) ≡
th A − th B
1 −th A th B
(b) sh 2 A ≡ 2 sh A ch A
4. (a) sh (A + B) ≡ sh A ch B + ch A sh B
(b)
sh
2 x + ch
2 x − 1
2ch 2 x coth 2 x
≡ tanh
4 x
5. Given Pe x − Qe −x ≡ 6 ch x − 2 sh x, find P
and Q
[P = 2, Q =−4]
6. If 5e x − 4e −x ≡ A sh x + B ch x, find A and B.
[A = 9, B = 1]
5.4 Solving equations involving
hyperbolic functions
Equations such as sinh x = 3.25 or coth x = 3.478 may
be determined using a calculator. This is demonstrated
in Worked Problems 15 to 21.
Problem 15. Solve the equation sh x = 3, correct
to 4 significant figures.
If sinh x = 3, then x = sinh −1 3
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 sh x = 3 is: x = 1.818, correct to 4
significant figures.
Problem 16. Solve the equation ch x = 1.52,
correct to 3 decimal places.
Using a calculator with a similar procedure as 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 = cosh x =
1.52. Hence, x = ±0.980
Problem 17. Solve the equation tanh θ = 0.256,
correct to 4 significant figures.
Using a calculator with a similar procedure as in Worked
Problem 15, check that gives
θ = 0.2618, correct to 4 significant figures.
Problem 18. Solve the equation sech x = 0.4562,
correct to 3 decimal places.
If sech x = 0.4562, then
x = sech
−1 0.4562 =
cosh −1
1
0.4562
since cosh =
1
sech
i.e. x = 1.421, correct to 3 decimal places.
With reference to the graph of y = sech x in Fig. 5.4, it
can be seen that there will be two values corresponding
to y = sech x = 0.4562
Hence, x = ±1.421
Problem 19. Solve the equation
cosech y = −0.4458, correct to 4 significant figures.
If cosech y = − 0.4458, then y = 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 coth A = 2.431,
correct to 3 decimal places.
If
coth A = 2.431,
then
A = coth −1 2.431 =
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 of y when x is 25, and (b) the
value of x when y = 54.30
Now try the following exercise
Exercise 21 Further problems on
hyperbolic identities
In Problems 1 to 4, prove the given identities.
1. (a) ch (P − Q) ≡ ch P ch Q − sh P sh Q
(b) ch 2x ≡ ch 2 x + sh 2 x
2. (a) coth x ≡ 2 cosech 2x + th x
(b) ch 2θ − 1 ≡2 sh 2 θ
3. (a) th (A − B) ≡
th A − th B
1 −th A th B
(b) sh 2 A ≡ 2 sh A ch A
4. (a) sh (A + B) ≡ sh A ch B + ch A sh B
(b)
sh
2 x + ch
2 x − 1
2ch 2 x coth 2 x
≡ tanh
4 x
5. Given Pe x − Qe −x ≡ 6 ch x − 2 sh x, find P
and Q
[P = 2, Q =−4]
6. If 5e x − 4e −x ≡ A sh x + B ch x, find A and B.
[A = 9, B = 1]
5.4 Solving equations involving
hyperbolic functions
Equations such as sinh x = 3.25 or coth x = 3.478 may
be determined using a calculator. This is demonstrated
in Worked Problems 15 to 21.
Problem 15. Solve the equation sh x = 3, correct
to 4 significant figures.
If sinh x = 3, then x = sinh −1 3
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 sh x = 3 is: x = 1.818, correct to 4
significant figures.
Problem 16. Solve the equation ch x = 1.52,
correct to 3 decimal places.
Using a calculator with a similar procedure as 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 = cosh x =
1.52. Hence, x = ±0.980
Problem 17. Solve the equation tanh θ = 0.256,
correct to 4 significant figures.
Using a calculator with a similar procedure as in Worked
Problem 15, check that gives
θ = 0.2618, correct to 4 significant figures.
Problem 18. Solve the equation sech x = 0.4562,
correct to 3 decimal places.
If sech x = 0.4562, then
x = sech
−1 0.4562 =
cosh −1
1
0.4562
since cosh =
1
sech
i.e. x = 1.421, correct to 3 decimal places.
With reference to the graph of y = sech x in Fig. 5.4, it
can be seen that there will be two values corresponding
to y = sech x = 0.4562
Hence, x = ±1.421
Problem 19. Solve the equation
cosech y = −0.4458, correct to 4 significant figures.
If cosech y = − 0.4458, then y = 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 coth A = 2.431,
correct to 3 decimal places.
If
coth A = 2.431,
then
A = coth −1 2.431 =
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 of y when x is 25, and (b) the
value of x when y = 54.30