Hyperbolic Cosecant

The Hyperbolic Cosecant calculator computes the hyperbolic cosecant function of a real number.

INSTRUCTIONS: Enter the following:

• (x) domain

Hyperbolic Cosecant ( csch ): The results is returned as a real number.

The Math / Science

The Hyperbolic Cosecant function, denoted as csch(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 tangent (tanh), hyperbolic cosecant (csch), hyperbolic secant (sech), and hyperbolic cotangent (coth).

The hyperbolic cosine function is defined as:

csch(x)= 2 / (e^x -  e^-x) 

Where:

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

Graphically, the csch(x) function resembles a mirrored version of the sinh(x) curve, with vertical asymptotes at x = 0.

The hyperbolic cosecant function appears in various mathematical and scientific contexts, including differential equations, geometry, signal processing, and physics, where it describes phenomena with exponential growth or decay.

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.