In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola.

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$\text{tanh} (x + k\pi i) = \text{tanh}\ x$ $\text{coth} (x + k\pi i) =\text{coth} x$ Relationship between inverse hyperbolic and inverse trigonometric functions sin -1 (ix) = isinh -1 x

Hyperbolic tangent "tanh" (pronounced "than"): tanh(x) = sinh(x) cosh(x) = e x − e-x e x + e-x. tanh vs tan . Hyperbolic cotangent: coth(x) = cosh(x) sinh(x) = e x + e-x e x − e-x . Hyperbolic secant: sech(x) = 1 cosh(x) = 2 e x + e-x . Hyperbolic cosecant "csch" or "cosech": csch(x) = 1 sinh(x) = 2 e x − e-x. Why the Word "Hyperbolic" ?

There are six hyperbolic functions are sinh x, cosh x, tanh x, coth x, sech x, csch x. In this article, we will define these hyperbolic functions and their properties, graphs, identities, derivatives, etc. along with some solved examples.

We know the identity $\tanh(ix)=i \tanh(x)$. My question: is it true that $\tanh^{-1}(ix) = i \tanh^{-1}(x)$ ? If not then is there a similar identity for arctangents?

Jun 24, 2024 · Integral Hyperbolic functions are used to solve various mathematical problems. The integral of hyperbolic cos x is hyperbolic sin x, i.e. The table added below shows the integration of various hyperbolic functions. Learn, Integration.

Calculate the relation between \(\coth\) function and \(\cot\) function by using the tan hyperbolic property \(tanx=-i\tanh{ix}\).

The hyperbolic functions are periodic w.r.t. the imaginary component with period 2πi for sinh, cosh, sech & cosech; and with period πi for tanh and coth. z representing a complex number, the functions sinh z and cosh z are said to be holomorphic or analytic (see below figures).

The hyperbolic tangent function is a function f: R → R is defined by f(x) = [e x – e-x] / [e x + e-x] and it is denoted by tanh x. tanh x = [e x – e-x] / [e x + e-x] Graph : y = tanh x. Properties of Hyperbolic Functions. The properties of hyperbolic functions are analogous to the trigonometric functions. Some of them are: Sinh (-x) = -sinh x