Hyperbolic functions

What hyperbolic functions covers

The exponential relatives of sine and cosine: the same identities with some signs changed, the shape of a hanging chain, and a new family of integrals. Core Pure content, examinable on both compulsory papers of 9FM0. Two lessons, and both of them start from the exponential definitions rather than from quoted results.

The main ideas

• sinh, cosh and tanh defined from exponentials, with their reciprocals, graphs, domains and ranges.
• The identity linking cosh squared and sinh squared, proved from the definitions in two lines.
• Osborn's rule, which converts a trigonometric identity by flipping the sign wherever a product of two sinh terms appears.
• The logarithmic forms of arsinh, arcosh and artanh, with the domain each one needs.
• Hyperbolic equations reduced to a quadratic in one hyperbolic function before anything is solved.
• Differentiating sinh, cosh and tanh, where no minus sign appears.
• Integrating to arsinh and arcosh, by recognition or by a hyperbolic substitution.

The results it turns on

cosh x = (e^x + e^−x)/2, and sinh x = (e^x − e^−x)/2

the definitions everything in the unit follows from

cosh²x − sinh²x = 1

the hyperbolic Pythagorean identity

arsinh x = ln(x + √(x² + 1)), and arcosh x = ln(x + √(x² − 1)) for x ≥ 1

the logarithmic forms an exact answer is written in

the derivative of sinh x is cosh x, and of cosh x is sinh x

the derivatives, with no sign change in either

∫1/√(x² + a²) gives arsinh(x/a), and ∫1/√(x² − a²) gives arcosh(x/a)

the two new standard integrals

cosh 2x = 2cosh²x − 1

the double angle result used to integrate cosh² x

Every result the hyperbolic functions notes display is collected, with its conditions, on the hyperbolic functions formulae page.

Where it usually goes wrong

• The two functions behave differently as equations. cosh x = k has two solutions for k > 1 and none below 1, while sinh x = k has one solution for any real k, so say which case you are in.
• The sign inside the root sorts the integral: x² + a² goes to arsinh, x² − a² to arcosh, and a² − x² to arcsin from Further calculus.
• Osborn's rule changes the sign only where a product of two sinh terms is present, including where one is hidden inside a tanh squared.

Where to start

Definitions and identities first, and derive rather than quote, since that derivation is the expected working in a show-that question. Calculus second. Revise the exponential and logarithm laws beforehand, and take the unit alongside Further calculus so that the three root forms are learned together.