Hyperbolic Tangent

The Hyperbolic Tangent calculator computes the hyperbolic tangent function of a real number.

INSTRUCTIONS: Enter the following:

• (x) domain

Hyperbolic Tangent ( tanh ): The results is returned as a real number.

The Math / Science

The Hyperbolic Tangent function, denoted as tanh(x), is a mathematical function defined for all real numbers x. It is one of the hyperbolic trigonometric functions, along with hyperbolic sine (sinsh), hyperbolic cosine (cosh), hyperbolic cosecant (csch), hyperbolic secant (sech), and hyperbolic cotangent (coth).

The hyperbolic tangent function is defined as:

tanh(x)= (e^x - e^-x) / (e^x + e^-x)

Where:

• e is the base of the natural logarithm, approximately equal to 2.71828.

Graphically, the tanh(x) function resembles an S-shaped curve that ranges from -1 to 1 as x varies from negative to positive infinity.

The hyperbolic tangent function shares certain properties with the regular tangent function, such as periodicity, but it is not periodic in the same way as the trigonometric tangent function. Instead, it is asymptotic to -1 as x approaches negative infinity and asymptotic to 1 as x approaches positive infinity.

The hyperbolic tangent function appears in various mathematical and scientific contexts, including differential equations, geometry, signal processing, and physics. It is particularly useful in areas where exponential growth or decay is involved, as well as in modeling phenomena with sigmoidal behavior.

Hyperbolic Functions:

• sinh: Hyperbolic Sine
  • `sinh(x) = (e^x - e^(-x))/2`
• cosh: Hyperbolic cosine
  • `cosh(x) = (e^x + e^(-x))/2`
• tanh: Hyperbolic tangent
  • `tanh(x) = (sinh(x))/(cosh(x)) = ((e^x - e^(-x))) / ((e^x + e^(-x)))`
• coth: Hyperbolic cotangent
  • `coth(x) = 1 / (tanh(x)) = ((e^x + e^(-x))) / ((e^x - e^(-x)))`
• csch: Hyperbolic cosecant
  • `csch(x) = 1 / (sinh(x)) = 2/(e^x - e^(-x))`
• sech: Hyperbolic secant
  • `sech(x) = 1 / (cosh(x)) = 2/(e^x + e^(-x))`

Inverse Hyperbolic Functions:

• arshinh: Arc Hyperbolic Sine
  • `arsinh(x) = ln(x+sqrt(x^2+1))` For: `-infty < x < infty`
• arcosh: Arc Hyperbolic Cosine
  • `arcosh(x) = ln(x+sqrt(x^2-1))` For: `-infty < x < infty`
• artanh: Arc Hyperbolic Tangent
  • `artanh(x) = 1/2 ln((1+x)/(1-x))` For: `-1 < x < 1` 
• arcsch: Arc Hyperbolic Cosecant
  • `arcsch(x) = ln(1/x + sqrt(1/x^2 +1))` For: `-infty < x < infty, xne0`
• arsech: Arc Hyperbolic Secant
  • `arsech(x) = ln(1/x + sqrt(1/x^2 - 1))` For: `0 <x le 1`
• arcoth: Arc Hyperbolic Cotangent
  • `arcoth(x) = 1/2 * ln( (x+1)/(x-1))` For: `-infty < x < -1 "or" 1 < x < infty`

The Science

Hyperbolic trigonometric functions are a family of mathematical functions closely related to ordinary trigonometric functions. While ordinary trigonometric functions (like sine, cosine, and tangent) are defined based on the unit circle, hyperbolic trigonometric functions are defined based on the geometry of the hyperbola. These functions have properties similar to their ordinary trigonometric counterparts. For example, sinh(x) and cosh(x) are analogs of sine and cosine, respectively, and have similar symmetries and periodic properties. However, instead of describing the relationships between angles and sides of right triangles, hyperbolic trigonometric functions describe the relationships between sides and diagonals of hyperbolic triangles. They appear in various mathematical contexts, including differential equations, complex analysis, and geometry, as well as in physics and engineering.