Higher Engineering Mathematics, Sixth Edition

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

1. 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.

4. sh2θ−shθ≡θ+

7

6

θ^3 +

31

120

θ^5

5. 2sh

θ

2

−ch

θ

2

≡− 1 +θ−

θ^2

8

+

θ^3

24

−

θ^4

384

+

θ^5

1920