Hyperbolic Trigonometry

While standard circular trigonometric functions are related to the coordinates of a unit circle (x2+y2=1x^2 + y^2 = 1), hyperbolic trigonometric functions are related to the coordinates of a standard unit hyperbola (x2−y2=1x^2 - y^2 = 1). These functions arise frequently in engineering and physics, such as the shape of a hanging cable (catenary) and special relativity.

Geometric Interpretation: Hyperbolic Sector Area

In circular trigonometry, the parameter θ\theta in cos⁡(θ)\cos(\theta) and sin⁡(θ)\sin(\theta) represents an angle (or arc length). However, in hyperbolic trigonometry, the parameter aa in cosh⁡(a)\cosh(a) and sinh⁡(a)\sinh(a) represents an area, not an angle.
For a standard unit hyperbola (x2−y2=1x^2 - y^2 = 1), if we draw a ray from the origin to a point (x,y)=(cosh⁡a,sinh⁡a)(x, y) = (\cosh a, \sinh a) on the right branch of the hyperbola, the area of the hyperbolic sector bounded by the x-axis, the ray, and the hyperbola is exactly a/2a/2.
This is perfectly analogous to the unit circle (x2+y2=1x^2 + y^2 = 1), where the area of a circular sector bounded by the x-axis and a point (cos⁡θ,sin⁡θ)(\cos \theta, \sin \theta) is exactly θ/2\theta/2. This profound geometric symmetry is why they are called "hyperbolic" functions.

Definitions of Hyperbolic Functions

Hyperbolic functions are defined using the exponential function exe^x.

Primary Hyperbolic Functions

The primary hyperbolic functions are hyperbolic sine (sinh), hyperbolic cosine (cosh), and hyperbolic tangent (tanh).
  • Hyperbolic Sine (sinh⁡x\sinh x): ex−e−x2\displaystyle \frac{e^x - e^{-x}}{2}
  • Hyperbolic Cosine (cosh⁡x\cosh x): ex+e−x2\displaystyle \frac{e^x + e^{-x}}{2}
  • Hyperbolic Tangent (tanh⁡x\tanh x): sinh⁡xcosh⁡x=ex−e−xex+e−x\displaystyle \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}
Reciprocal Functions
The remaining three functions are the reciprocals:
  • Hyperbolic Cosecant (csch x\text{csch } x): 1sinh⁡x=2ex−e−x\displaystyle \frac{1}{\sinh x} = \frac{2}{e^x - e^{-x}}
  • Hyperbolic Secant (sech x\text{sech } x): 1cosh⁡x=2ex+e−x\displaystyle \frac{1}{\cosh x} = \frac{2}{e^x + e^{-x}}
  • Hyperbolic Cotangent (coth⁡x\coth x): 1tanh⁡x=ex+e−xex−e−x\displaystyle \frac{1}{\tanh x} = \frac{e^x + e^{-x}}{e^x - e^{-x}}
Key Takeaways
  • Exponential Basis: Hyperbolic functions are linear combinations of exe^x and e−xe^{-x}.
  • Even and Odd: cosh⁡(x)\cosh(x) is an even function (cosh⁡(−x)=cosh⁡(x)\cosh(-x) = \cosh(x)), while sinh⁡(x)\sinh(x) and tanh⁡(x)\tanh(x) are odd functions.

Hyperbolic Identities

Hyperbolic identities closely resemble circular trigonometric identities, often with a difference in sign.

Fundamental Hyperbolic Identity

Just as cos⁡2x+sin⁡2x=1\cos^2 x + \sin^2 x = 1 is the fundamental identity of circular trigonometry (based on the unit circle), the fundamental hyperbolic identity is based on the unit hyperbola (x2−y2=1x^2 - y^2 = 1).

Fundamental Hyperbolic Identity

The core relationship between hyperbolic cosine and sine.

Variables

SymbolDescriptionUnit
xxArgument (real number or angle)-

Fundamental Hyperbolic Identity

By dividing this identity by cosh⁡2x\cosh^2 x or sinh⁡2x\sinh^2 x, we obtain the related identities:

Hyperbolic Tangent and Secant Identity

Derived by dividing the primary identity by hyperbolic cosine squared.

Variables

SymbolDescriptionUnit
xxArgument-

Hyperbolic Cotangent and Cosecant Identity

Derived by dividing the primary identity by hyperbolic sine squared.

Variables

SymbolDescriptionUnit
xxArgument-

Addition and Subtraction Formulas

The sum and difference formulas for hyperbolic functions are:

Hyperbolic Sum and Difference Formulas

Formulas for the sum and difference of arguments in hyperbolic functions.

Variables

SymbolDescriptionUnit
x,yx, yArguments-
Key Takeaways
  • Osborn's Rule: You can convert any circular trigonometric identity involving only sines and cosines to a hyperbolic identity by replacing cos⁡\cos with cosh⁡\cosh and sin⁡\sin with sinh⁡\sinh, and changing the sign of any term containing a product of two sines (or implied sines like tan⁡2\tan^2).

Relationship with Circular Trigonometry

Because of Euler's Formula (eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos \theta + i \sin \theta), hyperbolic functions (defined by exponentials) and circular trigonometric functions are intimately connected through complex numbers.
By substituting an imaginary argument ixix into the circular functions, we obtain the hyperbolic functions:
  • sin⁡(ix)=isinh⁡(x)\sin(ix) = i \sinh(x)
  • cos⁡(ix)=cosh⁡(x)\cos(ix) = \cosh(x)
  • tan⁡(ix)=itanh⁡(x)\tan(ix) = i \tanh(x)
Conversely, substituting an imaginary argument into hyperbolic functions yields circular functions:
  • sinh⁡(ix)=isin⁡(x)\sinh(ix) = i \sin(x)
  • cosh⁡(ix)=cos⁡(x)\cosh(ix) = \cos(x)
  • tanh⁡(ix)=itan⁡(x)\tanh(ix) = i \tan(x)
This profound symmetry bridges the geometry of the circle and the hyperbola.

Inverse Hyperbolic Functions

Because hyperbolic functions are defined in terms of exe^x, their inverses can be expressed in terms of natural logarithms.

Logarithmic Definitions

The inverse hyperbolic functions are defined as follows:

Inverse Hyperbolic Logarithmic Forms

Defines inverse hyperbolic functions explicitly in terms of natural logarithms.

Variables

SymbolDescriptionUnit
xxValue within the domain of the respective inverse function-
ln⁡\lnNatural logarithm (base e)-
Key Takeaways
  • Inverses as Logarithms: Inverse hyperbolic functions provide a direct link to natural logarithms, which is incredibly useful for integrating certain rational functions and radicals.
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