Sinh Calculator (Hyperbolic Sine)

Understanding Hyperbolic Sine

The hyperbolic sine function, denoted sinh(x), is defined as (e^x - e^-x) / 2. Unlike the ordinary sine function which relates to circles, hyperbolic functions relate to hyperbolas. The hyperbolic sine is an odd function, meaning sinh(-x) = -sinh(x), and its domain and range are both all real numbers.

Hyperbolic Function Formulas

sinh(x)

The hyperbolic sine, defined using the exponential function.

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

cosh(x)

The hyperbolic cosine, the even counterpart of sinh.

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

tanh(x)

The hyperbolic tangent, the ratio of sinh to cosh.

tanh(x) = sinh(x) / cosh(x)

Key Identity

The hyperbolic analog of the Pythagorean identity.

cosh^2(x) - sinh^2(x) = 1

Inverse (asinh)

The inverse hyperbolic sine function.

asinh(x) = ln(x + sqrt(x^2 + 1))

Derivative

The derivative of sinh(x) is cosh(x).

d/dx sinh(x) = cosh(x)

Applications

Hyperbolic functions appear in many areas of mathematics and physics. The catenary curve (shape of a hanging chain) is described by cosh(x). Hyperbolic functions are used in special relativity for Lorentz transformations, in solving certain differential equations, and in engineering for describing transmission line behavior and heat transfer.

Key Properties

sinh(0) = 0, and sinh is an odd function: sinh(-x) = -sinh(x).
sinh(x) grows exponentially for large |x|, approximately e^x/2.
The derivative of sinh(x) is cosh(x), and vice versa.
sinh is the imaginary part: sinh(x) = -i sin(ix).