acosh(): The Inverse Hyperbolic Cosine

acosh() returns the inverse hyperbolic cosine of a value. It undoes cosh() on its principal real branch, making it useful in hyperbolic geometry, catenary curves, and formulas involving exponential growth. Unlike circular cosine, hyperbolic cosine is built from exponentials rather than rotation around a circle.

Valid real inputs

For real arithmetic, acosh(x) is defined for x greater than or equal to 1 and returns a non-negative value. Thus acosh(1) is 0. Values below 1 have no real result, although they can be evaluated in complex arithmetic. Check constraints early, especially when x is computed from measured data.

One equivalent form is acosh(x) = log(x + sqrt(x² − 1)). This relation explains why values near 1 require care: subtraction and square roots can magnify rounding effects. Use sqrt() and log() for an explicit formula when you need to inspect intermediate values, but prefer acosh for a direct, readable expression.

Practical use

Hyperbolic functions model hanging cables, relativistic transformations, and solutions to differential equations. To verify a result, apply cosh(acosh(x)); within numerical precision it returns x. Related functions include asinh() and atanh(), whose domains differ significantly.

Do not confuse acosh with acos. Their similar names refer to different families: acos works with circular angles and accepts real values from −1 to 1, while acosh works with hyperbolic values and accepts real values starting at 1.