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 a quotation.

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

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.