Hyperbolic Functions
Hyperbolic functions are nearly similar to trigonometric functions. They are defined through the algebraic expressions which include exponential function ex and its inverse function e-x, where e is the Euler’s constant .The approximate value of e is 2.718281.The basic hyperbolic functions are hyperbolic sine (sinh) and hyperbolic cosine (cosh), from which the hyperbolic tangent (tanh) function is derived. The other functions coth,sech,cosech are reciprocals of the above three functions correspondingly. The definitions for the basic hyperbolic functions are
1. The function f: R->R defined by f(x) = (ex-e-x)/2 is called hyperbolic
sine function and it is denoted by Sinh x.
Sinh x= (ex-e-x)/2
2. The function f: R->R defined by f(x) = (ex+e-x)/2 is called hyperbolic
cosine function and it is denoted by cosh x.
cosh x= (ex+e-x)/2
3. The function f:R->R defined by f(x)=(ex-e-x)/(ex+e-x) is called hyperbolic
tangent function and it is denoted by tanh x.
tanh x=(ex-e-x)/(ex+e-x)
4. The function f: R->R defined by f(x) = (ex+e-x)/(ex-e-x) is called hyperbolic
cotangent function and it is denoted by coth x.
coth x=(ex+e-x)/(ex-e- x))
5. The function f: R->R defined by f(x) = 2/(ex+e-x) is called hyperbolic
secant function and it is denoted by sech x.
sech x=2/(ex+e-x)
6. The function f: R->R defined by f(x) = 2/(ex-e-x) is called hyperbolic
cosecant function and it is denoted by cosech x.
cosech x=2/(ex-e-x)
The properties of the hyperbolic functions are analogous to trigonometric functions. Let us discuss in detail,
sinh(-x) = (?sinhx)
cosh(-x) = coshx
The derivatives of the basic hyperbolic functions are
d/dx sinh(x) = coshx
d/dx cosh(x) = sinhx
The relation of hyperbolic functions to trigonometric functions is as follows:
sinhx =(? i sin(ix))
coshx= cos(ix)
tanhx=?i tan(ix)
The hyperbolic identities as similar to that of trigonometric functions are
cosh2(x) - sinh2(x) = 1
tanh2(x) + sech2(x) = 1
coth2(x) - cosech2(x) = 1
Hyperbolic functions are related to hyperbola, in the same way, as the trigonometric functions are related to circle.
A unit hyperbola can be defined with the help of the two basic hyperbolic functions as follows:
x2 - y2 = 1 where x = cosht , y = sinht ; -? < t < ?
x2 - y2 = cosh2 t - sinh2 t =1
The main use of these functions is to integrate common and simple functions with less computation and the other use of these functions can be observed in the models of real life problems.
Below are the exam,ples based on hyperbolic functions -
Solve cosh2x –sinh2x
`[(e^x+e^(-x))/(2)]^2 -[(e^x -e^(-x))/(2)]^2`
= [(e2x +e-2x+2*ex * e-x)/4] – [(e2x +e-2x– 2*ex * e-x)/4]
= `(4e^xe^(-x))/(4)`
=`(4*1)/(4)` =1
Exercise:
1. Solve tanh2x+sech2X
2. Solve cosech2x-coth2x
3. Express hyperbolic tangent of 5
4. express hyperblic cosine of 8
