Hyperbolic Functions

What are Hyperbolic Functions?

In Mathematics, hyperbolic functions are similar to trigonometric or circular functions. Typically, hyperbolic functions are defined by algebraic expressions that include exponential function (ex) and its inverse exponential function (e-x), where e is the Euler constant. Here, we 're going to discuss the basic hyperbolic functions, hyperbolic functions formulas, their properties, identities, and examples in detail. Hyperbolic functions are named in the same way as trigonometric functions with the letter 'h' added to each name.

Hyperbolic Function Definition

Certain even and odd combinations of the exponential functions ex and e-x arise so frequently in engineering, Physics, and Mathematics that they're given special names. Hyperbolic functions are analogous trigonometric functions in that they are named the same as trigonometric functions with the letter 'h' appended to each name. These special functions have the same relationship to the hyperbola that trigonometric functions have to the circle. For this reason, they are collectively known as hyperbolic functions and are individually called hyperbolic sine, hyperbolic cosine, and so on. In addition to modeling, they can be used as solutions to some types of partial differential equations.

Hyperbolic functions are analogs of a circular function or a trigonometric function. Hyperbolic motion arises in the solutions of linear differential equations, the measurement of distance, and the angles in hyperbolic geometry, the Laplace equations in the Cartesian coordinates. Generally, the hyperbolic function takes place in the particular statement called the hyperbolic angle. Generally, the hyperbolic function takes place in the real argument called the hyperbolic angle. The basic hyperbolic functions are:

  • Hyperbolic sine (sinh)

  • Hyperbolic cosine (cosh)

  • Hyperbolic tangent (tanh)

Hyperbolic Functions Formulas

The two major basic hyperbolic functions are "sinh" and "cosh"

Hyperbolic Sine Function Formula -

sinh(x) = 

ex − e−x

2

(pronounced "shine")

Hyperbolic Cosine -

cosh(x) = 

ex + e−x

2

(pronounced "cosh")

Identities

  • sinh(−x) = −sinh(x)

  • cosh(−x) = cosh(x)

And

  • tanh(−x) = −tanh(x)

  • coth(−x) = −coth(x)

  • sech(−x) = sech(x)

  • cosech(−x) = −cosech(x)

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 address in detail the basic hyperbolic functions, diagrams, properties, and inverse hyperbolic functions.

Differentiation of Hyperbolic Functions

Hyperbolic Functions Formulas - 

The basic hyperbolic functions formulas along with its graph functions 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 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

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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

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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)

Hyperbolic Function Identities

Properties of hyperbolic trigonometric functions

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)

Inverse Hyperbolic Functions

The inverse function of hyperbolic functions is 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 of the complex plane is regarded 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 Example

Hyperbolic functions examples solutions

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

FAQ (Frequently Asked Questions)

1. What do you Mean by Hyperbolic Functions?

Solution - An angle function expressed as the relationship between the distances from the point of the hyperbola to the origin and the coordinate axes, as hyperbolic sine or hyperbolic cosine: often represented as combinations of exponential functions.

Hyperbolic functions are defined in terms of exponentials, and definitions lead to properties such as the extension and differentiation of hyperbolic functions as infinite series. They are written as trig functions cosine (cos), sine (sin), tangent (tan), but they have a 'h' at the end.