How the Sinh (Hyperbolic Sine) Function Works
The hyperbolic sine function sinh(x) is defined in terms of the natural exponential function: sinh(x) = (eˣ − e⁻ˣ) / 2. Unlike the ordinary trigonometric sine, which is periodic and bounded between −1 and 1, sinh(x) is unbounded and strictly increasing — it can take on any real value as x ranges over all real numbers. This calculator evaluates sinh(x) for any real x, along with the related hyperbolic cosine cosh(x) = (eˣ + e⁻ˣ) / 2 and hyperbolic tangent tanh(x) = sinh(x) / cosh(x).
Formula and method
Enter any real number x. Note that x is not an angle in degrees or radians — sinh, cosh, and tanh are hyperbolic functions built directly from the exponential function, not from a circle. The calculator computes eˣ and e⁻ˣ, subtracts them, and divides by 2 to get sinh(x). It also reports cosh(x) and tanh(x) for the same x, and internally checks the identity cosh²x − sinh²x = 1 to confirm the result is numerically consistent.
Key properties
- Odd function: sinh(−x) = −sinh(x), so the graph passes through the origin and is symmetric about it (rotate 180° and it looks the same).
- Domain and range: sinh(x) is defined for every real x, and its output covers every real number — unlike sin(x), which never leaves [−1, 1].
- Small-x approximation: near x = 0, sinh(x) ≈ x + x³/6 + ⋯, so for very small |x|, sinh(x) is approximately equal to x itself.
- Large-x approximation: for large positive x, e⁻ˣ becomes negligible, so sinh(x) ≈ eˣ/2; for large negative x, eˣ becomes negligible, so sinh(x) ≈ −e⁻ˣ/2.
Common uses
Hyperbolic functions show up whenever a system's growth is exponential rather than circular. The catenary curve traced by a hanging cable or chain is y = a·cosh(x/a); sinh and its inverse (arcsinh) appear in solutions to certain differential equations, in special relativity (rapidity addition), and throughout hyperbolic geometry. Because cosh²x − sinh²x = 1, the pair (cosh θ, sinh θ) parametrizes a hyperbola the same way (cos θ, sin θ) parametrizes a circle.