` = cosh( "x" )`

Enter a value for all fields

The Hyperbolic Cosine calculator computes the hyperbolic cosine function of a real number.

INSTRUCTIONS: Enter the following:

• (x) domain

Hyperbolic Cosine ( cosh ): The results is returned as a real number.

The Math / Science

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

The hyperbolic cosine function is defined as:

cosh(x)= (e^x + e^-x) / 2

Where:

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

Graphically, the cosh(x) function resembles a symmetric exponential curve that approaches positive infinity as x tends toward positive or negative infinity.

The hyperbolic cosine function shares many properties with the regular cosine function, such as being an even function (cosh(-x) = cosh(x)), having a range of [1, ∞), and being related to the exponential function. It appears in various mathematical and scientific contexts, including differential equations, geometry, signal processing, and physics.

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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.