082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

2.4 Review of some common engineering functions and techniques 103
Inorder tosolve this equation wefirst multiplyboth sides by e x , togive
0 =e x e x −e x 2y+e x e −x
Using the laws of indices,
0=e 2x −2e x y+1
We can now write this as a quadratic equation in e x and solve it using the standard
formula.This gives
(e x ) 2 −2y(e x )+1=0
e x = −b ± √ b 2 −4ac
2a
= −(−2y) ± √ (−2y) 2 −4(1)(1)
2(1)
= 2y ± √ 4y 2 −4
=y ± √ y
2
2 −1
Asx 0,the function e x 1 so wereject the negative rootand itfollows that
e x =y+ √ y 2 −1
from which we get
(
x=ln y + √ )
y 2 −1
Recall thaty =coshxand sox = cosh −1 y. Therefore
(
cosh −1 y = ln y + √ )
y 2 −1
Hencetheequationforthepairofwirescanalsobewrittenintermsoflogarithmsas
πε
C = [ 0
ln (d/2r) + √ ]
(d/2r) 2 −1
EXERCISES2.4.5
A word of warning about the inverse of coshx is needed. The calculator returns a
value of 1.9827 for cosh −1 (3.7). Note, however, that cosh(−1.9827) = 3.7. The value
−1.9827 is not returned by the calculator; only values on [0,∞) will be returned. This
isbecause the domainofy = coshx isrestrictedtoensureithas aninverse function.
1 Evaluate the following:
(a) sinh3 (b) cosh1.6
(c) tanh0.95 (d) sech1
(e) cosech2 (f) coth1.5
(g) cosh(−3) (h) cosech(−1.6)
(i) sinh(−2) (j) coth(−2.7)
(k) tanh(−1.4) (l) sech(−0.5)
2 Evaluate
(a) sinh −1 3
(b) cosh −1 2
(c) tanh −1 (−0.25)
3 Express
(a) 6e x +5e −x in terms ofsinhx andcoshx,
(b) 4e 2x −3e −2x in terms ofsinh2x andcosh2x,
(c) 2e −3x −5e 3x in terms ofsinh3x andcosh3x.
4 Express
(a) 4sinhx +3coshx in terms ofe x ande −x ,
(b) 3sinh2x −cosh2x in termsofe 2x ande −2x ,
(c) 3cosh3x −0.5sinh3x in terms ofe 3x ande −3x .
5 Expressae x +be −x ,whereaandbare constants,in
terms ofcoshx andsinhx.