Sinh Calculator — Hyperbolic Sine & Related Functions
Compute sinh(x) = (eˣ − e⁻ˣ)/2 and the full set of hyperbolic functions — cosh, tanh, coth — for any real x, with the curve plotted and the fundamental identity verified.
Hyperbolic sine — (eˣ − e⁻ˣ) / 2
- 1
eˣ
e^(1) = 2,718282 - 2
e⁻ˣ
e^(−1) = 0,367879 - 3
eˣ − e⁻ˣ
2,718282 − 0,367879 = 2,350402 - 4
sinh(x) = (eˣ − e⁻ˣ) ÷ 2
2,350402 ÷ 2 = 1,175201
Jak działa ten kalkulator?
sinh(x) = (eˣ − e⁻ˣ)/2. Enter x to get sinh, cosh, tanh, and coth, plus a verification that cosh²−sinh²=1. Unlike circular sine, sinh is unbounded and strictly increasing. Used in catenary mechanics, special relativity, and integration.
Wzór
How this is calculated
The hyperbolic sine sinh(x) = (eˣ − e⁻ˣ)/2 is an odd, strictly increasing function that grows without bound in both directions and passes through the origin with slope 1. Its companion cosh(x) = (eˣ + e⁻ˣ)/2 is always ≥ 1, even, and forms the shape of a catenary (a hanging chain). Together they satisfy the fundamental identity cosh²(x) − sinh²(x) = 1, the hyperbolic analogue of the Pythagorean identity.
The identity check in the stats grid should show 1.000000 for any x; tiny deviations (e.g. 1.0000000000000002) are normal double-precision floating-point rounding. tanh(x) = sinh/cosh is bounded between −1 and +1 (approaching ±1 as x → ±∞). coth(x) = cosh/sinh is undefined at x = 0 and approaches ±1 for large |x|.
Hyperbolic functions appear in: the catenary curve (shape of a hanging cable or chain), special relativity (rapidity w = tanh⁻¹(v/c)), heat-transfer fins, and many calculus antiderivatives such as ∫ 1/√(1+x²) dx = sinh⁻¹(x).
Najczęściej zadawane pytania
sin(x) is periodic with range [−1, 1], defined via the unit circle. sinh(x) is not periodic, grows to ±∞, and is defined via the unit hyperbola as (eˣ − e⁻ˣ)/2. Both are odd functions (f(−x) = −f(x)).
Computers store numbers in binary floating point with ~15 significant digits. Squaring and subtracting two large numbers loses a few of those digits, so the result may differ from 1 by up to 10⁻¹⁴. This is expected and not a bug.
The catenary equation y = a·cosh(x/a) describes the shape of any uniform hanging cable — used in bridge and power-line design. sinh appears in the analytic solution to relativistic kinematics and in the closed-form antiderivatives needed in many physics and engineering integrals.
Znany również jako
TG we-Calculate Editorial Team. (2026). Sinh Calculator — Hyperbolic Sine & Related Functions [Online calculator]. TG we-Calculate. https://we-calculate.com/pl/calculator/sinh-calculator
TG we-Calculate Editorial Team. "Sinh Calculator — Hyperbolic Sine & Related Functions." TG we-Calculate. 2026. https://we-calculate.com/pl/calculator/sinh-calculator.
TG we-Calculate Editorial Team, "Sinh Calculator — Hyperbolic Sine & Related Functions," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/pl/calculator/sinh-calculator
@misc{wecalculate_sinh_calculator, title = {Sinh Calculator — Hyperbolic Sine & Related Functions}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/pl/calculator/sinh-calculator}}, year = {2026}, note = {TG we-Calculate} }
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