Page 44 - Standard Handbook Of Petroleum & Natural Gas Engineering

Trigonometry 33

Hyperbolic Functions
The hyperbolic sine, hyperbolic cosine, etc. of any number x are functions related
to the exponential function ex. Their definitions and properties are very similar
to the trigonometric functions and are given in Table 1-5.
The inverse hyperbolic functions, sinh-’x, etc., are related to the logarithmic
functions and are particularly useful in integral calculus. These relationships may
be defined for real numbers x and y as

sinh-’ (x/y) = In( x + Jx* + y2 ) - In y

ash-’ (x/y) = In( x + Jx* - y2 ) - In y

tanh-’(x/y) = 1/2 In[(y + x)/(y - x)]

coth-’(x/y) = 1/2 In[(x + y)/(x - y)]

Table 1-5
Hyperbolic Functions
sinh x = 1/2(ex - e-.)
cosh x = 1/2(eK + e-.)
tanh x = sinh x/cosh x
csch x = l/sinh x
sech x = llcosh x
coth x = l/tanh x
sinh(-x) = -sinh x
cosh(-x) = COSh x
tanh(-x) = -tanh x
cosh’x - sinh’x = 1
1 - tanh‘x = sech’x
1 - COth’X = - CSCh’X
sinh(x f y) = sinh x cosh y f cosh x sinh y
cosh(x f y) = cosh x cosh y f sinh x sinh y
tanh(x f y) = (tanh x f tanh y)/(l f tanh x tanh y)
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh2x + sinhzx
tanh 2x = (2 tanh x)/(l + tanh2x)
sinh(d2) = .\11/2(cosh x - 1)
~0~h(x/2) ,/1/2(cosh x + 1)
=
tanh(d2) = (cosh x - l)/(sinh x) = (sinh x)/(cosh x + 1)