Work out inverse hyperbolic functions instantly with clear inputs, formula shown and shareable results.
Each inverse hyperbolic function has a closed logarithmic form: arsinh x = ln(x + √(x² + 1)) for all x, arcosh x = ln(x + √(x² − 1)) for x ≥ 1, and artanh x = ½ln((1 + x)/(1 − x)) for |x| < 1.
arsinh
arsinh x = ln(x + √(x² + 1))
arcosh
arcosh x = ln(x + √(x² − 1)), x ≥ 1
artanh
artanh x = ½·ln((1 + x) / (1 − x)), |x| < 1
ln(2 + √5) ≈ 1.4436355, and sinh of that returns 2.
Because cosh never dips below 1, so no real input produces a smaller output.