Hyperbolic Functions

In Mathematics, the hyperbolic functions are similar to the trigonometric functions or circular functions. Generally, the hyperbolic functions are defined through the algebraic expressions that include the exponential function (ex) and its inverse exponential functions (e-x), where e is the Euler’s constant. The basic hyperbolic functions are:

  • Hyperbolic sine (sinh)
  • Hyperbolic cosine (cosh)
  • Hyperbolic tangent (tanh)

From these three basic functions, the other functions such as hyperbolic cosecant (cosech), hyperbolic secant(sech) and hyperbolic cotangent (coth) functions are derived. Let us discuss the basic hyperbolic functions, graphs, properties, and inverse hyperbolic functions in detail.

Hyperbolic Functions Graphs

The basic hyperbolic functions and their graphs are given below:

Hyperbolic Sine Function

The hyperbolic sine function is a function f: R → R is defined by f(x) = [ex– e-x]/2 and it is denoted by sinh x

Sinh x = [ex– e-x]/2

Graph : y = Sinh x

Hyperbolic Sine Function

Hyperbolic Cosine Function

The hyperbolic cosine function is a function f: R → R is defined by f(x) = [ex +e-x]/2 and it is denoted by cosh x

cosh x = [ex + e-x]/2

Graph : y = cosh x

Hyperbolic Cos Function

Hyperbolic Tangent Function

The hyperbolic tangent function is a function f: R → R is defined by f(x) = [ex – e-x] / [ex + e-x] and it is denoted by tanh x

tanh x = [ex – e-x] / [ex + e-x]

Graph : y = tanh x

Hyperbolic Tan Function

Properties of Hyperbolic Functions

The properties of hyperbolic functions are analogous to the trigonometric functions. Some of them are:

  1. Sinh (-x) = -sinh x
  2. Cosh (-x) = cosh x
  3. Sinh 2x = 2 sinh x cosh x
  4. Cosh 2x = cosh 2x + sinh 2x

The derivatives of hyperbolic functions are:

  1. d/dx sinh (x) = cosh x
  2. d/dx cosh (x) = sinh x

Some relations of hyperbolic function to the trigonometric function are as follows:

  1. Sinh x = – i sin(ix)
  2. Cosh x = cos (ix)
  3. Tanh x = -i tan(ix)

Inverse Hyperbolic Functions

The inverse function of hyperbolic functions are known as inverse hyperbolic functions. It is also known as area hyperbolic function. The inverse hyperbolic functions provides the hyperbolic angles corresponding to the given value of the hyperbolic function. Those functions are denoted by sinh-1, cosh-1, tanh-1, csch-1, sech-1, and coth-1. The inverse hyperbolic function in complex plane is defined as follows:

  • Sinh-1 x = ln(x + √[1+x2])
  • Cosh-1 x = ln(x + √[x2-1])
  • Tanh-1 x = (½)[ln(1+x) – ln(1-x)

Hyperbolic Function Identities

The hyperbolic function identities are similar to the trigonometric functions. Some identities are:

Pythagorean Trigonometric Identities

  • Cosh2 (x) – sinh2 (x) = 1
  • tanh2 (x) + sech2 (x) = 1
  • Coth2 (x) – cosech2 (x) = 1

Sum to Product

  • sinh x + sinh y = 2 sinh( (x+y)/2) cosh((x-y)/2)
  • sinh x – sinh y = 2cosh((x+y)/2) sinh((x-y)/2)
  • cosh x + cosh y = 2cosh((x+y)/2) cosh((x-y)/2)
  • cosh x – cosh y = 2 sinh((x+y)/2) sinh((x-y)/2)

Product to Sum

  • 2sinh x cosh y = sinh(x + y) + sinh(x -y)
  • 2cosh x sinh y = sinh(x + y) – sinh(x – y)
  • 2sinh x sinh y = cosh(x + y) – cosh(x – y)
  • 2cosh x cosh y = cosh(x + y) + cosh(x – y).

Sum and Difference Identities

  • sinh(x ± y) = sinh x cosh x ± coshx sinh y
  • cosh(x ±y) = cosh x cosh y ± sinh x sinh y
  • tanh(x ±y) = (tanh x ± tanh y) / (1± tanh x tanh y )
  • coth(x ±y) = (coth x coth y ± 1) / (coth y ±coth x)

Hyperbolic Function Example

Example: Solve cosh2 x – sinh2 x

Solution:

Given: cosh2 x – sinh2 x

We know that

Sinh x = [ex– e-x]/2

cosh x = [ex + e-x]/2

cosh2 x – sinh2 x = [ [ex + e-x]/2 ]2 – [ [ex – e-x]/2 ]2

cosh2 x – sinh2 x = (4ex-x) /4

cosh2 x – sinh2 x = (4e0) /4

cosh2 x – sinh2 x = 4(1) /4 = 1

Therefore, cosh2 x – sinh2 x = 1

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