Hyperbolic Sine

Online calculator for calculating the hyperbolic sine of an angle

Sinh Calculator

Universal Domain

The Sinh(x) or hyperbolic sine shows S-curve behavior for all real numbers.

Enter any angle value in degrees or radians
Results
Sinh:

Sinh Function Curve

Curve of the Sinh function

The Sinh function shows S-curve behavior passing through origin with exponential growth.
Domain: ℝ, Range: ℝ

S-Curve Behavior of Sinh

The hyperbolic sine function exhibits characteristic S-curve properties:

  • Domain: ℝ (all real numbers)
  • Range: ℝ (all real numbers)
  • Zero Point: Sinh(0) = 0
  • Symmetry: Odd function Sinh(-x) = -Sinh(x)
  • Growth: Exponential for large |x|
  • Shape: Classic S-curve through origin

Exponential Representation of Sinh Function

The hyperbolic sine function is expressed through exponential functions:

Basic Formula
\[\text{Sinh}(x) = \frac{e^x - e^{-x}}{2}\]

Difference of exponentials divided by 2

Growth Relation
\[\text{Sinh}(x) \approx \frac{e^x}{2} \text{ for large } x\]

Exponential growth for large arguments

Formulas for the Sinh Function

Definition
\[\text{Sinh}(x) = \frac{e^x - e^{-x}}{2}\]

Exponential difference for all x ∈ ℝ

Hyperbolic Identity
\[\cosh^2(x) - \sinh^2(x) = 1\]

Fundamental hyperbolic identity

Derivative
\[\frac{d}{dx} \text{Sinh}(x) = \text{Cosh}(x)\]

Derivative is hyperbolic cosine

Symmetry Property
\[\text{Sinh}(-x) = -\text{Sinh}(x)\]

Odd function (antisymmetric)

Addition Formula
\[\text{Sinh}(x + y) = \text{Sinh}(x)\text{Cosh}(y) + \text{Cosh}(x)\text{Sinh}(y)\]

Addition formula for hyperbolic sine

Special Values

Important Values
Sinh(0) = 0 Sinh(1) ≈ 1.175 Sinh(ln(2)) = 0.75
Universal Domain
\[x \in \mathbb{R} \text{ (all real numbers)}\]

Function defined everywhere

Zero Point
\[\text{Sinh}(0) = 0\]

Function passes through origin

Properties
  • Odd function
  • S-curve shape
  • Exponential growth
  • Passes through origin
Growth Pattern
\[\text{Sinh}(x) \approx \frac{e^{|x|}}{2} \cdot \text{sgn}(x)\]

Exponential growth for large arguments

Applications

Relativity theory, engineering calculations, wave equations, mathematical modeling.

Detailed Description of the Sinh Function

Definition and Input

The hyperbolic sine function Sinh(x) is defined as the difference of exponential functions. It exhibits characteristic S-curve behavior passing through the origin with exponential growth for large arguments.

Key Advantage: The Sinh function accepts any real number and passes through origin!
Input

The angle is given in degrees (full circle = 360°) or radians (full circle = 2π). The unit of measurement used is set using the Degrees or Radians menu. All real numbers are valid inputs.

Result

The result can be any real number. For large positive arguments the function grows exponentially, for large negative arguments it approaches negative infinity exponentially.

Using the Calculator

Enter any angle value. The Sinh function calculates the hyperbolic sine, which represents the S-curve growth pattern.

Mathematical Properties

Function Properties
  • Domain: ℝ (all real numbers)
  • Range: ℝ (all real numbers)
  • Zero Point: Sinh(0) = 0
  • Symmetry: Odd function Sinh(-x) = -Sinh(x)
S-Curve Properties
  • Passes through origin (0, 0)
  • Monotonically increasing function
  • Exponential growth for large |x|
  • S-shaped curve with inflection point at origin
Applications
  • Special Relativity: Rapidity calculations
  • Engineering: Catenary and suspension problems
  • Wave Equations: Hyperbolic partial differential equations
  • Mathematical Analysis: Complex function theory
Practical Notes
  • No domain restrictions: accepts any real number
  • Origin passage: Sinh(0) = 0
  • Odd function: Sinh(-x) = -Sinh(x)
  • S-curve: Characteristic exponential growth pattern

Calculation Examples

Small Values

Sinh(0) = 0

Sinh(0.5) ≈ 0.521

Sinh(1) ≈ 1.175

Negative Values

Sinh(-0.5) ≈ -0.521

Sinh(-1) ≈ -1.175

Sinh(-2) ≈ -3.627

Large Values

Sinh(5) ≈ 74.20

Sinh(10) ≈ 11013

Exponential growth

Physics and Engineering Applications

Special Relativity

Rapidity:

v = c · tanh(φ), γ = cosh(φ)

Hyperbolic motion in spacetime

Application: Particle acceleration and relativistic kinematics.

Wave Equations

Hyperbolic PDEs:

Wave propagation solutions

Vibration and oscillation analysis

Example: String vibrations and membrane oscillations.

Important Mathematical Relationships

Hyperbolic Identity
\[\cosh^2(x) - \sinh^2(x) = 1\] \[\sinh(x + y) = \sinh(x)\cosh(y) + \cosh(x)\sinh(y)\]

Fundamental Identity: Analogous to trigonometric identity.

Calculus Properties
\[\frac{d}{dx}\sinh(x) = \cosh(x)\] \[\int \sinh(x) dx = \cosh(x) + C\]

Derivative: Derivative is hyperbolic cosine.


IT Functions

Decimal, Hex, Bin, Octal conversion • Shift bits left or right • Set a bit • Clear a bit • Bitwise AND • Bitwise OR • Bitwise exclusive OR

Special functions

Airy • Derivative Airy • Bessel-I • Bessel-Ie • Bessel-J • Bessel-Je • Bessel-K • Bessel-Ke • Bessel-Y • Bessel-Ye • Spherical-Bessel-J • Spherical-Bessel-Y • Hankel • Beta • Incomplete Beta • Incomplete Inverse Beta • Binomial Coefficient • Binomial Coefficient Logarithm • Erf • Erfc • Erfi • Erfci • Fibonacci • Fibonacci Tabelle • Gamma • Inverse Gamma • Log Gamma • Digamma • Trigamma • Logit • Sigmoid • Derivative Sigmoid • Softsign • Derivative Softsign • Softmax • Struve • Struve table • Modified Struve • Modified Struve table • Riemann Zeta

Hyperbolic functions

ACosh • ACoth • ACsch • ASech • ASinh • ATanh • Cosh • Coth • Csch • Sech • Sinh • Tanh

Trigonometrische Funktionen

ACos • ACot • ACsc • ASec • ASin • ATan • Cos • Cot • Csc • Sec • Sin • Sinc • Tan • Degree to Radian • Radian to Degree