Hyperbolic Cosine Calculator

Definition

The hyperbolic cosine is the real function cosh : ℝ → [1, +∞) given by cosh(x) = (eˣ + e⁻ˣ)/2, where eˣ = exp(x) is the exponential. Equivalently cosh(x) = cos(i x) in the analytic continuation, but this page evaluates only the real case. The global minimum is cosh(0) = 1; the function is even: cosh(−x) = cosh(x).

The exponential definition is standard in the NIST Digital Library of Mathematical Functions, §4.28–4.35 (Hyperbolic Functions), and in Abramowitz and Stegun, Chapter 4. MathWorld (Hyperbolic Cosine) lists the same identities. The calculator uses cosh when available, or else (eˣ + e⁻ˣ)/2 directly.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Historical and conceptual context

Hyperbolic functions were introduced in the eighteenth century by Vincenzo Riccati and Johann Heinrich Lambert in connection with the hyperbola x² − y² = 1, by analogy with circular trigonometry. The parametrization (cosh t, sinh t) traces the right branch of the unit hyperbola, just as (cos t, sin t) traces the circle. The name “hyperbolic cosine” stresses that analogy, not periodicity — cosh is not periodic on ℝ.

In physics cosh appears in the hanging cable (catenary y = a cosh(x/a)), in special relativity (rapidities), and in wave-equation solutions. Conceptually cosh(x) is not a “cosine of an angle” in degrees — the argument is dimensionless (or in nepers), not an angular degree measure.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Formal properties

Domain: all of ℝ. Codomain: [1, +∞). Continuity and C∞ smoothness. Derivative: (cosh x)′ = sinh x, where sinh(x) = (eˣ − e⁻ˣ)/2. Second derivative: (cosh x)″ = cosh x, so cosh solves y″ = y with y(0) = 1, y′(0) = 0.

Hyperbola identity: cosh²(x) − sinh²(x) = 1 for every real x (NIST DLMF §4.35). Addition: cosh(x + y) = cosh x cosh y + sinh x sinh y. Evenness: cosh(−x) = cosh(x). Taylor series: cosh x = Σ_{n=0}^∞ x^{2n}/(2n)!.

The inverse on [1, +∞) is arcosh(y) = ln(y + √(y² − 1)). This calculator computes only the forward cosh map, not arcosh.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Full derivation

Step 1. From the exp series: eˣ = Σ xⁿ/n!, e⁻ˣ = Σ (−x)ⁿ/n!. Summing the even terms yields cosh x = Σ x^{2n}/(2n)!.

Step 2. (eˣ + e⁻ˣ)/2 at x = 0 gives (1 + 1)/2 = 1. Differentiating: d/dx[(eˣ + e⁻ˣ)/2] = (eˣ − e⁻ˣ)/2 = sinh x.

Step 3 (identity). cosh² − sinh² = [(eˣ + e⁻ˣ)² − (eˣ − e⁻ˣ)²]/4 = 4/4 = 1 after expansion.

Step 4 (circle vs hyperbola). For cos and sin one has cos² + sin² = 1 (circle); for cosh and sinh: cosh² − sinh² = 1 (hyperbola). The sign pattern is systematic in DLMF §4.28.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Special cases

Values: cosh(0) = 1; cosh(1) ≈ 1.54308; cosh(ln 2) = (2 + 1/2)/2 = 1.25; cosh(−2) = cosh(2) ≈ 3.7622. For large |x|, cosh(x) ∼ (1/2) e^{|x|}.

When |x| exceeds about 710 in floating-point arithmetic, eˣ overflows to Infinity and cosh loses a finite result — the calculator then reports an overflow error.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Numerical and computational notes

The browser evaluates cosh(x) (floating-point arithmetic). The panel shows eˣ, e⁻ˣ, sinh(x), and the check cosh² − sinh² ≈ 1. Comma and period are equivalent. Degree/radian switches do not apply — the argument is not a trigonometric angle.

For moderate x the formula (eˣ + e⁻ˣ)/2 is stable; for very large x a library cosh is preferable (avoiding direct overflow of both exponentials). This calculator still warns on overflow.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Related concepts and authorities

Related: the exponential eˣ (building block of the definition), natural logarithm (inverse of exp), and cosine of an angle (trigonometric analogy, different geometry).

Authorities: NIST DLMF §4.28–4.35; Abramowitz & Stegun, Chapter 4; MathWorld — Hyperbolic Cosine; Wikipedia — Hyperbolic functions. The calculator does not replace an analysis course.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.