48 NUMBER AND ALGEBRA(b) Wheny= 54 .30, 54. 30 =40 chx
40, from whichchx
40=54. 30
40= 1. 3575Following the above procedure:(i)ex(^40) +e
−x
40
2
= 1. 3575
(ii) e
x
(^40) +e
−x
(^40) = 2 .715, i.e. e
x
(^40) +e
−x
(^40) − 2. 715 = 0
(iii) (e
x
(^40) )^2 + 1 − 2 .715e
x
(^40) = 0
i.e. (e
x
(^40) )^2 − 2 .715e
x
(^40) + 1 = 0
(iv) e
x
(^40) =
−(− 2 .715)±
√
[(− 2 .715)^2 −4(1)(1)]
2(1)
715 ±
√
(3.3712)
2
715 ± 1. 8361
2
Hence e
x
(^40) = 2 .2756 or 0.43945
(v)
x
40
=ln 2.2756 or
x
40
=ln(0.43945)
Hence
x
40
= 0 .8222 or
x
40
=−0.8222
Hencex=40(0.8222) orx=40(−0.8222);
i.e.x=±32.89, correct to 4 significant figures.
Now try the following exercise.
Exercise 26 Further problems on hyper-
bolic equations
In Problems 1 to 5 solve the given equations
correct to 4 decimal places.
shx= 1 [0.8814]
2 chx=3[±0.9624]
3.5shx+ 2 .5chx=0[−0.8959]
2 shx+3chx= 5 [0.6389 or−2.2484]
4 thx− 1 = 0 [0.2554]
A chain hangs so that its shape is of the form
y=56 ch (x/56). Determine, correct to 4 sig-
nificant figures, (a) the value ofywhenxis
35, and (b) the value ofxwhen[ yis 62.35.
(a) 67. 30
(b) 26. 42
]5.5 Series expansions for coshxand
sinhxBy definition,ex= 1 +x+x^2
2!+x^3
3!+x^4
4!+x^5
5!+···from Chapter 4.
Replacingxby−xgives:e−x= 1 −x+x^2
2!−x^3
3!+x^4
4!−x^5
5!+···.coshx=1
2(ex+e−x)=1
2[(1 +x+x^2
2!+x^3
3!+x^4
4!+x^5
5!+···)+(1 −x+x^2
2!−x^3
3!+x^4
4!−x^5
5!+···)]=1
2[(
2 +2 x^2
2!+2 x^4
4!+···)]i.e.coshx= 1 +x^2
2!+x^4
4!+···(which is valid for
all values ofx). coshxis an even function and
contains only even powers ofxin its expansionsinhx=1
2(ex−e−x)=1
2[(1 +x+x^2
2!+x^3
3!+x^4
4!+x^5
5!+···)−(1 −x+x^2
2!−x^3
3!+x^4
4!−x^5
5!+···)]=1
2[2 x+2 x^3
3!+2 x^5
5!+ ···]i.e.sinhx=x+x^3
3!+x^5
5!+···(which is valid for
all values ofx). sinhxis an odd function and contains
only odd powers ofxin its series expansionProblem 15. Using the series expansion for
chxevaluate ch 1 correct to 4 decimal place.chx= 1 +x^2
2!+x^4
4!+ ···from above