Hyperbolic Functions

Hyperbolic functions are similar to trigonometric functions, but instead of unit circles, they are defined using rectangular hyperbolas. In trigonometry, the coordinates on a unit circle are represented as (cos θ, sin θ), whereas in hyperbolic functions, the pair (cosh θ, sinh θ) represents points on the right half of an equilateral hyperbola. 

They are used in solving linear differential equations, hyperbolic geometry, and Laplace’s equations in Cartesian coordinates.

The three basic hyperbolic functions are:

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

Hyperbolic functions are expressed through exponential function e^x and its inverse e^-x (here, e = Euler’s constant). 

Hyperbolic Sine (sinh)

The hyperbolic sine function is a function f: ℝ → ℝ such that f(x) = sinh x, which is expressed as

sinh x = ${\dfrac{e^{x}-e^{-x}}{2}}$

Now, let us graph the function y = sinh x and see how it behaves.

Hyperbolic Cosine (cosh)

The hyperbolic cosine function is a function f: ℝ → ℝ such that f(x) = cosh x, which is expressed as

cosh x = ${\dfrac{e^{x}+e^{-x}}{2}}$ 

Now, on graphing the hyperbolic cosine function y = cosh x, we get:

Hyperbolic Tangent (tanh)

The hyperbolic tangent function is a function f: ℝ → ℝ such that f(x) = tanh x, which is expressed as

tanh x = ${\dfrac{e^{x}-e^{-x}}{e^{x}+e^{-x}}}$

Now, on graphing the hyperbolic tangent function y = tanh x, we get

Other Hyperbolic Functions

In addition to the three basic hyperbolic functions, there are three other hyperbolic functions equivalent to sine, cosine, and tangent. They are secant, cosecant, and cotangent:

Hyperbolic Cosecant (cosech)

The hyperbolic cosecant function (denoted by cosech or csch) is the reciprocal of the hyperbolic sine function, which is given by the expression: 

cosech x = ${\dfrac{1}{\sinh x}}$ = ${\dfrac{2}{e^{x}-e^{-x}}}$

Now, on graphing the hyperbolic cosecant function y = cosech x, we get

Hyperbolic Secant (sech)

Similarly, the hyperbolic secant function (sech) is the reciprocal of the hyperbolic cosine function, which is given by the expression:

sech x = ${\dfrac{1}{\cosh x}}$ = ${\dfrac{2}{e^{x}+e^{-x}}}$

Now, on graphing the hyperbolic secant function y = sech x, we get

Hyperbolic Cotangent (coth)

The hyperbolic cotangent function (coth) is the reciprocal of the hyperbolic tangent function, which is given by the expression:

coth x = ${\dfrac{1}{\tanh x}}$ = ${\dfrac{e^{x}+e^{-x}}{e^{x}-e^{-x}}}$

Now, on graphing the hyperbolic cotangent function y = coth x, we get

Here is the summary of all graphs of the hyperbolic functions:

Domain and Range

The domain and range of the six hyperbolic functions are listed:

Hyperbolic Function	Domain	Range
sinh x	(-∞, ∞)	(-∞, ∞)
cosh x	(-∞, ∞)	[1, ∞)
tanh x	(-∞, ∞)	(-1, 1)
cosech x	(-∞, 0) ∪ (0, ∞)	(-∞, 0) ∪ (0, ∞)
sech x	(-∞, ∞)	(0, 1]
coth x	(-∞, 0) ∪ (0, ∞)	(-∞, – 1) ∪ (1, ∞)

Identities

Hyperbolic identities, similar to those of trigonometric functions, are related to the relationships between hyperbolic functions. These are universally true for all values of the variables involved.

Pythagorean Trigonometric Identities

• cosh² (x) – sinh² (x) = 1
• tanh² (x) + sech² (x) = 1
• coth² (x) – cosech² (x) = 1

Even-Odd Identities

• sinh (-x) = -sinh (x)
• cosh (−x) = cosh (x)
• tanh (−x) = −tanh (x)
• coth (-x) = -coth x
• sech (-x) = sech x
• cosec (-x) = -cosech x

Sum to Product

• sinh x + sinh y = ${2\sinh \left( \dfrac{x+y}{2}\right) \cosh \left( \dfrac{x-y}{2}\right)}$
• sinh x – sinh y = ${2\cosh \left( \dfrac{x+y}{2}\right) \sinh \left( \dfrac{x-y}{2}\right)}$
• cosh x + cosh y = ${2\cosh \left( \dfrac{x+y}{2}\right) \cosh \left( \dfrac{x-y}{2}\right)}$
• cosh x – cosh y = ${2\sinh \left( \dfrac{x+y}{2}\right) \sinh \left( \dfrac{x-y}{2}\right)}$

Product to Sum

• 2 sinh x cosh y = sinh (x + y) + sinh (x -y)
• 2 cosh x sinh y = sinh (x + y) – sinh (x – y)
• 2 sinh x sinh y = cosh (x + y) – cosh (x – y)
• 2 cosh 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) = ${\dfrac{\tanh x\pm \tanh y}{1\pm \tanh x\tanh y}}$
• coth (x ± y) = ${\dfrac{\coth x\coth y\pm 1}{\coth y\pm \coth x}}$

Double Angle Identities

• cosh 2x = 1 + 2 sinh2 x = 2 cosh2 x – 1
• cosh 2x = cosh2 x + sinh2 x
• sinh 2x = 2 ⋅ sinh x ⋅ cosh x

Derivatives

The derivatives of hyperbolic functions are similar to those of trigonometric functions.

• ${\dfrac{d}{dx}\left( \sinh x\right) =\cosh x}$
• ${\dfrac{d}{dx}\left( \cosh x\right) =\sinh x}$
• ${\dfrac{d}{dx}\left( \tanh x\right) =sech ^{2}x}$
• ${\dfrac{d}{dx}\left( cosech \  x\right) =-cosech \  x\cdot \coth x}$
• ${\dfrac{d}{dx}\left( sech \  x\right) =-sech \  x\cdot \tanh x}$
• ${\dfrac{d}{dx}\left( \coth x\right) =-cosech ^{2}x}$

Inverse Hyperbolic Functions

The inverse hyperbolic functions are the inverse operations of the hyperbolic functions. These functions are also known as area hyperbolic functions that provide the hyperbolic angles corresponding to a given value of the hyperbolic function. 

Here is the list of all inverse hyperbolic functions in the complex plane:

• sinh^-1 x = ${\ln \left( x+\sqrt{1+x^{2}}\right)}$
• cosh^-1 x = ${\ln \left( x+\sqrt{x^{2}-1}\right)}$
• tanh^-1 x = ${\dfrac{1}{2}\ln \left( \dfrac{1+x}{1-x}\right)}$
• cosech^-1 x = ${\ln \left( \dfrac{1+\sqrt{1+x^{2}}}{x}\right)}$
• sech^-1 x = ${\ln \left( \dfrac{1+\sqrt{1-x^{2}}}{x}\right)}$
• coth^-1 x = ${\dfrac{1}{2}\ln \left( \dfrac{x+1}{x-1}\right)}$

Solved Examples