Hyperbolic functions and their inverses

Learning objectives

Define sinh x = ½(eˣ − e⁻ˣ), cosh x = ½(eˣ + e⁻ˣ) and tanh x = sinh x / cosh x, sketch their graphs, and state the domain and range of each, noting that cosh x ≥ 1 for all real x

Prove and use the identities cosh²x − sinh²x = 1, 1 − tanh²x = sech²x and coth²x − 1 = cosech²x, and the addition and double-angle formulae for sinh and cosh, watching the sign changes compared with the circular identities

Derive the logarithmic forms of the inverse hyperbolic functions, arsinh x = ln(x + √(x² + 1)) for all real x, arcosh x = ln(x + √(x² − 1)) for x ≥ 1 and artanh x = ½ln((1 + x)/(1 − x)) for |x| less than 1, stating the restriction that makes arcosh single-valued

Solve equations involving hyperbolic functions, including those that reduce to a quadratic in eˣ, and reject any root that gives a negative value of eˣ or lies outside the domain

Differentiate and integrate hyperbolic and inverse hyperbolic functions, and recognise the standard integrals of 1/√(x² + a²) and 1/√(x² − a²) as giving arsinh and arcosh forms