Hyperbolic Sine

The Hyperbolic Sine calculator computes the hyperbolic sine function of a real number.

INSTRUCTIONS: Enter the following:

• (x) domain

Hyperbolic Sine ( sinh ): The results is returned as a real number.

The Math / Science

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

The hyperbolic sine function is defined as:

sinh(x)= (e^x - e^-x) / 2

Where:

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

The graph of sinh(x) resembles a symmetric exponential curve, asymptotically approaching positive and negative infinity as x moves away from zero.

The hyperbolic sine function arises naturally in various areas of mathematics, physics, and engineering, including differential equations, complex analysis, signal processing, and special relativity. It shares many properties with the ordinary sine function but is related to the geometry of hyperbolas instead of circles.

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.