Hyperbolic Cosine Calculator (cosh)

cosh(x) = (eˣ + e⁻ˣ) / 2. Catenary curve: y = a · cosh(x/a).

Inputs

Result

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General calculation reads

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How to use this calculator

  • Enter x.
  • Read cosh(x) + companion sinh, tanh.

About this calculator

Hyperbolic cosine: cosh(x) = (eˣ + e⁻ˣ) / 2. Even function: cosh(−x) = cosh(x). Always ≥ 1 (minimum at x = 0). The catenary curve y = a·cosh(x/a) is the natural shape of a chain or cable hanging under gravity — and is famously different from a parabola, even though they look similar. Used in architecture (St. Louis Gateway Arch), suspension cable design.

Frequently asked

Catenary vs. parabola?+
Both look like U shapes, but catenary y = cosh(x) has different curvature. The Gateway Arch is an inverted (weighted) catenary.
When is cosh(x) = 1?+
Only at x = 0. cosh(0) = (1 + 1) / 2 = 1. It's the minimum.
Hyperbolic vs. circular geometry?+
Circular trig parameterizes the unit circle. Hyperbolic trig parameterizes the unit hyperbola x² − y² = 1.
Inverse?+
cosh⁻¹(x) = ln(x + √(x² − 1)) for x ≥ 1. Two values otherwise (positive + negative).
Rapidly grows?+
For large x, cosh(x) ≈ eˣ / 2. Same exponential growth as sinh.

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