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hyperbolic tangent " tanh" (/ ˈ t æ ŋ, ˈ t æ n tʃ, ˈ θ æ n /), [5] hyperbolic cotangent " coth" (/ ˈ k ɒ θ, ˈ k oʊ θ /), [6] [7] hyperbolic secant " sech" (/ ˈ s ɛ tʃ, ˈ ʃ ɛ k /), [8] hyperbolic cosecant " csch" or "cosech" (/ ˈ k oʊ s ɛ tʃ, ˈ k oʊ ʃ ɛ k / [3]) corresponding to the derived trigonometric functions ...
Graphs of the inverse hyperbolic functions The hyperbolic functions sinh, cosh, and tanh with respect to a unit hyperbola are analogous to circular functions sin, cos, tan with respect to a unit circle. The argument to the hyperbolic functions is a hyperbolic angle measure.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
Identity 1: + = The following two results follow from this and the ratio identities. To obtain the first, divide both sides of + = by ; for the second, divide by .
Twice the area of the purple triangle is the stereographic projection s = tan 1 / 2 ϕ = tanh 1 / 2 ψ. The blue point has coordinates (cosh ψ, sinh ψ). The red point has coordinates (cos ϕ, sin ϕ). The purple point has coordinates (0, s). The integral of the hyperbolic secant function defines the Gudermannian function:
Twice the area of the purple triangle is the stereographic projection s = tan 1 / 2 ϕ = tanh 1 / 2 ψ. The blue point has coordinates (cosh ψ, sinh ψ). The red point has coordinates (cos ϕ, sin ϕ). The purple point has coordinates (0, s). Graph of the Gudermannian function. Graph of the inverse Gudermannian function.
The hyperbolic secant distribution shares many properties with the standard normal distribution: it is symmetric with unit variance and zero mean, median and mode, and its probability density function is proportional to its characteristic function.
To convert a trigonometric identity to the equivalent hyperbolic trigonometric identity, Osborn’s rule states to first write out all the cosine and sine compound angles terms to their expanded constituent parts. Then exchange all the cosine and sine terms to cosh and sinh terms.