Cosh Calculator — Hyperbolic Cosine
Compute the hyperbolic cosine cosh(x) = (eˣ + e⁻ˣ) / 2 for any real x, and see sinh(x), tanh(x), the fundamental identity cosh² − sinh² = 1, and a plotted curve.
Hyperbolic cosine — always ≥ 1
- 1
eˣ
e^1 = 2,718282Natural exponential of x. - 2
e⁻ˣ
e^(−1) = 0,367879 - 3
cosh(x) = (eˣ + e⁻ˣ) ÷ 2
(2,718282 + 0,367879) ÷ 2 = 1,543081
Cum funcționează acest calculator?
cosh(x) = (eˣ + e⁻ˣ) / 2. It is always ≥ 1, minimum at x = 0, grows exponentially for large |x|, and traces the catenary curve. The identity cosh²(x) − sinh²(x) = 1 mirrors sin² + cos² = 1 for circular functions. sinh(x) = (eˣ − e⁻ˣ) / 2 and tanh(x) = sinh/cosh is bounded in (−1, 1).
Formulă
How this is calculated
The hyperbolic functions are defined using the natural exponential eˣ rather than a unit circle. The hyperbolic cosine cosh(x) = (eˣ + e⁻ˣ) / 2 is the even component of eˣ — it reflects the average of eˣ and its mirror e⁻ˣ. Because both eˣ and e⁻ˣ are always positive, cosh(x) is always ≥ 1, reaching its minimum of 1 at x = 0, and growing without bound in both directions. This gives it the characteristic U-shaped curve known as a catenary — the shape a hanging chain or cable takes under gravity.
The companion function sinh(x) = (eˣ − e⁻ˣ) / 2 is the odd component of eˣ; it is always zero at x = 0 and grows as the difference between eˣ and e⁻ˣ. The hyperbolic tangent tanh(x) = sinh(x) / cosh(x) is bounded between −1 and 1, making it common as a smooth activation function in machine learning.
The fundamental identity cosh²(x) − sinh²(x) = 1 is the hyperbolic analogue of the circular identity sin² + cos² = 1. Unlike the circular functions, the hyperbolic functions are not periodic — they grow exponentially for large |x|. For very large x (e.g. x > 20), the e⁻ˣ term is negligible and cosh(x) ≈ eˣ / 2.
Întrebări frecvente
A flexible cable or chain hanging under its own weight takes the shape of a catenary, described exactly by y = a·cosh(x/a). The Gateway Arch in St Louis is a catenary arch. Hyperbolic functions also appear in special relativity (rapidity), the shape of soap films between rings, and signal processing.
No. They are related by substituting an imaginary argument: cosh(x) = cos(ix), where i = √−1. For real x, cosh(x) ≥ 1 and is unbounded, while cos(x) oscillates between −1 and 1. Their graphs look similar near x = 0 but diverge rapidly.
The inverse hyperbolic cosine is arccosh(x) = ln(x + √(x²−1)), defined for x ≥ 1. It is the positive branch of the inverse since cosh is not one-to-one on all reals (it is symmetric about x = 0).
Cunoscut și ca
TG we-Calculate Editorial Team. (2026). Cosh Calculator — Hyperbolic Cosine [Online calculator]. TG we-Calculate. https://we-calculate.com/ro/calculator/cosh-calculator
TG we-Calculate Editorial Team. "Cosh Calculator — Hyperbolic Cosine." TG we-Calculate. 2026. https://we-calculate.com/ro/calculator/cosh-calculator.
TG we-Calculate Editorial Team, "Cosh Calculator — Hyperbolic Cosine," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/ro/calculator/cosh-calculator
@misc{wecalculate_cosh_calculator, title = {Cosh Calculator — Hyperbolic Cosine}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/ro/calculator/cosh-calculator}}, year = {2026}, note = {TG we-Calculate} }
Te-a ajutat acest calculator?