50 Higher Engineering Mathematics
In the series expansion for shx,letx= 2 θ, then:
sh2θ= 2 θ+
( 2 θ)^3
3!
+
( 2 θ)^5
5!
+···
= 2 θ+
4
3
θ^3 +
4
15
θ^5 +···
Hence
ch
(
θ
2
)
−sh2θ=
(
2 +
θ^2
4
+
θ^4
192
+···
)
−
(
2 θ+
4
3
θ^3 +
4
15
θ^5 +···
)
= 2 − 2 θ+
θ^2
4
−
4
3
θ^3 +
θ^4
192
−
4
15
θ^5 +···as far the term inθ^5
Now try the following exercise
Exercise 23 Further problems on series
expansions for coshxand sinhx
- Use the series expansion for chxto evaluate,
correct to 4 decimal places: (a) ch 1.5(b)ch0. 8
[(a) 2.3524 (b) 1.3374]
2. Use the series expansion for shx to evalu-
ate, correct to 4 decimal places: (a) sh0. 5
(b) sh2
[(a) 0.5211 (b) 3.6269]
3. Expand the following as a power series as far
as the term inx^5 :(a)sh3x(b) ch2x
⎡
⎢
⎣
(a) 3x+
9
2
x^3 +
81
40
x^5
(b) 1+ 2 x^2 +
2
3
x^4
⎤
⎥
⎦
In Problems 4 and 5, prove the given identities,
the series being taken as far as the term inθ^5
only.
- sh2θ−shθ≡θ+
7
6
θ^3 +
31
120
θ^5
- 2sh
θ
2
−ch
θ
2
≡− 1 +θ−
θ^2
8
+
θ^3
24
−
θ^4
384
+
θ^5
1920