Knowra Inverse hyperbolic functions Inverse hyperbolic functions Inverse hyperbolic functions undo hyperbolic functions on specified domains and ranges. They can be expressed using logarithms and describe quantities such as the inverse hyperbolic sine and cosine.
Inverse hyperbolic sine : The inverse of the hyperbolic sine, defined for every real input as arsinh(x). Its logarithmic formula, ln(x + √(x² + 1)), illustrates the family’s standard construction.
Hyperbolic sine : The function sinh(x) = (eˣ − e⁻ˣ)/2. Its exponential definition leads directly to the quadratic equation used to derive arsinh.
Catenary : The curve formed by a perfectly flexible hanging chain under uniform gravity. Its equation uses cosh, while solving for horizontal position naturally introduces arcosh.
Inverse sine : The principal inverse of the circular sine function, with values in [−π/2, π/2]. Unlike arsinh, inverse sine has a bounded real domain and range.
Inverse hyperbolic cosine : The inverse of hyperbolic cosine on the nonnegative domain, defined for inputs at least one. Its restricted domain and logarithmic formula show why inverse hyperbolic functions need branch choices.
Hyperbolic cosine : The function cosh(x) = (eˣ + e⁻ˣ)/2. Its evenness makes its inverse single-valued only after restricting its domain.
Gudermannian function : A function relating circular angles to hyperbolic functions without using complex numbers. Inverse hyperbolic functions express the Gudermannian through logarithmic formulas.
Inverse cosine : The principal inverse of the circular cosine function, with values in [0, π]. Its real domain resembles arcosh’s, but its trigonometric geometry differs.
Inverse hyperbolic tangent : The inverse of hyperbolic tangent, defined for real inputs strictly between −1 and 1. Its formula, one-half ln((1+x)/(1−x)), makes its finite domain and boundary behavior explicit.
Hyperbolic tangent : The function tanh(x) = sinh(x)/cosh(x), with values strictly between −1 and 1. Its bounded range becomes the real domain of its inverse.
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