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

102 Chapter 2 Engineering functions
Inversehyperbolicfunctions
The inverse hyperbolic functions use the familiar notation. Both y = sinhx and y =
tanhx are one-to-one functions and no domain restriction is needed for an inverse to be
defined.However,on (−∞,∞),y = coshxisamany-to-onefunction.Ifthedomainis
restricted to [0,∞) the resulting function is one-to-one and an inverse function can be
defined.
The inverse of the function sinhx is denoted by sinh −1 x. Here the −1 must not be
interpreted as a power but rather the notation we use for the inverse function. Similarly
the inverses of coshx and tanhx aredenoted by cosh −1 x and tanh −1 x respectively.
Values of sinh −1 x, cosh −1 x and tanh −1 x can be obtained using a scientific
calculator.
Example2.21 Evaluate
(a) cosh −1 (3.7) (b) sinh −1 (−2) (c) tanh −1 (0.5)
Solution Using a calculator we get
(a) 1.9827 (b) −1.4436 (c) 0.5493
Engineeringapplication2.15
Capacitancebetweentwoparallelwires
Althoughitmaynotseemobvious,asmallcapacitanceexistsbetweentwowiresthat
run close to each other and at certain frequencies this can be a significant factor for
someelectrical systems.
Themutualcapacitancepermetre,C,betweentwolongparallelwiresinaireach
havingaradiusrmetresandwiththewirecentresseparatedbyd metresiscalculated
using
πε
C = 0
cosh −1 (d/2r)
Thisexpressionincludesaninversehyperbolicfunction.Intheequationd>2r,otherwise
the wires would be overlapping. The constant ε 0
is a fundamental physical
constant called the permittivity of free space. It has an approximate value of
8.85 ×10 −12 Fm −1 .
Recall the general expression forthe hyperbolic function coshx:
coshx = ex +e −x
2
To derive the inverse of this hyperbolic function, we need to restrict the domain to
[0, ∞), that isx 0.We lety = coshx and then solve forxintermsofy:
y = ex +e −x
2
2y=e x +e −x
0=e x −2y+e −x