Hyperbolic Functions

The hyperbolic cosine function is defined to be:

cosh(x) = (ex + e-x)/2

And the hyperbolic sine function is defined as:

sinh(x) = (ex - e-x)/2

From the above definition we can see the following properties of the cosh(x) and sinh(x) functions:
cosh(x) is an even function:

cosh(-x) = (e-x + ex)/2
         = (ex + e-x)/2
         = cosh(x)

Hence the function is even function.
Similarly the sinh(x) is an odd function:

sinh(-x) = (e-x - ex)/2
         = -((ex - e-x)/2)
         = -sinh(x)

Hence sinh(x) function is an odd function.
Range of cosh(x):

cosh(x) = (e-x + ex)/2

Let y = ex
cosh(x) = (y + 1/y)/2 . Now (y + 1/y) ≥ 2 for y>0.
Hence cosh(x) ≥ 1  

Therefore, range of cosh(x) is [1, ∞).

Range of sinh(x):

sinh(x) = (ex - e-x)/2
Let sinh(x) = y.
So y = (e2x - 1)/2ex
   2yex = e2x - 1
   e2x - 2yex + 1 = 0
   x = ln(y + √(y2 + 1)) 

Clearly since x is in (-∞, ∞), y has to lie in (-∞, ∞) so that the above log gives all real numbers.

Analogy between hyperbolic sines and cosines and normal trigonometric sines and cosines.

Osbourne’s Rule:
This rule states that the normal trigonometric identities in sines and cosines remain the same even with hyperbolic sines and cosines, i.e., cos(x) can be replaced by cosh(x) and sin(x) can be replaced by sinh(x).But it is to be remembered that when there is a multiplication of two sine functions there is a sign change in the identity.

For example,

sin(2A) = 2sin(A)cos(A)

It remains the same in hyperbolic functions, i.e.,

sinh(2x) = 2sinh(x)cosh(x)

cos(2A) = 1 - 2sin2(x)

which becomes,

cosh(2x) = 1 + 2sinh2(x)

There is also analogy in geometry. While (cos(t), sin(t)) represent the points on a unit circle, (cosh(x), sinh(x)) represent points on a hyperbola with x-intercept = 1.


cosh2(x) - sinh2(x) = 1
= L.H.S 
=  ((ex + e-x)/2)2 - ((ex - e-x)/2)2
= ((e2x + e-2x + 2)/4) - ((e2x + e-2x - 2)/4) 
= 4/4
= 1

Other Hyperbolic Functions:

tanh(x) = sinh(x)/cosh(x)
coth(x) = cosh(x)/sinh(x)
cosech(x) = 1/sinh(x)
sech(x) = 1/cosh(x)

Derivatives of Hyperbolic Functions:

= d(cosh(x))/dx 
= d((ex + e-x)/2)/dx
= 1/2(d(ex + e-x)/dx)
= 1/2(ex - e-x)
= sinh(x)


d(sinh(x))/dx = cosh(x)

You can see the difference between derivatives of sin(x) and cos(x) and sinh(x) and cosh(x). So you can carry out any operation for hyperbolic functions by just replacing them with their defined functions.

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