Hyperbolic functions 49
- 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
formy=56cosh
(x
56). Determine, correct to
4 significant figures, (a) the value ofywhen
xis 35, and (b) the value ofxwhenyis 62.35
[(a) 67.30 (b)± 26 .42]
5.5 Series expansions for coshxand sinhx
By 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 allvalues ofx). coshxis an even function and contains
only even powers ofxin its expansion.
sinhx=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 expansion.Problem 23. Using the series expansion for chx
evaluate ch1 correct to 4 decimal places.chx= 1 +x^2
2!+x^4
4!+···from aboveLet x= 1 ,then ch1= 1 +12
2 × 1+14
4 × 3 × 2 × 1+16
6 × 5 × 4 × 3 × 2 × 1+ ···= 1 + 0. 5 + 0. 04167 + 0. 001389 +···i.e. ch1= 1. 5431 , correct to 4 decimal places,
which may be checked by using a calculator.Problem 24. Determine, correct to 3 decimal
places, the value of sh 3 using the series expansion
for shx.shx=x+x^3
3!+x^5
5!+···from aboveLetx= 3 ,thensh3= 3 +33
3!
+35
5!
+37
7!
+39
9!
+311
11!
+···= 3 + 4. 5 + 2. 025 + 0. 43393 + 0. 05424+ 0. 00444 +···i.e. sh3= 10. 018 , correct to 3 decimal places.Problem 25. Determine the power series for
2ch(
θ
2)
−sh2θas far as the term inθ^5.In the series expansion for chx,letx=θ
2then:2ch(
θ
2)
= 2[
1 +(θ / 2 )^2
2!+(θ / 2 )^4
4!+···]= 2 +θ^2
4+θ^4
192+···