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.

$\tanh \frac{x}{2} = \pm \sqrt{\frac{\cosh x - 1}{\cosh x + 1}}$ [+ if x > 0, - if x . 0] $=\frac{\text{sinh}(x)}{1 + \text{cosh}(x)} = \frac{\text{cosh}(x) - 1}{\text{sinh}(x)}$ Multiple angle formulas

The two basic hyperbolic functions are "sinh" and "cosh": Hyperbolic Sine: sinh(x) = e x − e-x 2 (pronounced "shine") Hyperbolic Cosine: cosh(x) = e x + e-x 2 (pronounced "cosh") They use the natural exponential function e x. And are not the same as sin(x) and cos(x), but a little bit similar: sinh vs sin. cosh vs cos. Catenary

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.

Bài này sẽ nêu các công thức xác định sinh x, cosh x, tanh x, coth x, sech x, csch x và một số tính chất cơ bản liên quan. sinh, cosh, tan...

Sep 25, 2020 · sinh(-x) = -sinh(x); cosh(-x) = cosh(x); tanh(-x) = -tanh(x). Their ranges of values differ greatly from the corresponding circular functions: cosh(x) has its minimum value of 1 for x = 0, and tends to infinity as x tends to plus or minus infinity;

Nov 25, 2024 · Learn the different hyperbolic trigonometric functions, including sine, cosine, and tangent, with their formulas, examples, and diagrams. Also, learn their identities.

Hyperbolic Definitions sinh(x) = ( e x - e-x)/2 . csch(x) = 1/sinh(x) = 2/( e x - e-x) . cosh(x) = ( e x + e-x)/2 . sech(x) = 1/cosh(x) = 2/( e x + e-x) . tanh(x ...

The graphs and properties such as domain, range and asymptotes of the 6 hyperbolic functions: sinh(x), cosh(x), tanh(x), coth(x), sech(x) and csch(x) are presented. The six hyperbolic functions are defined as follows: Hyperbolic Sine Function : \( \sinh(x) = \dfrac{e^x - e^{-x}}{2} \)

Jul 28, 2023 · For example, the hyperbolic tangent (tanh) is defined as the ratio of the hyperbolic sine to the hyperbolic cosine: tanh(x) = sinh(x) / cosh(x). Similarly, the hyperbolic cotangent (coth) is the reciprocal of the hyperbolic tangent: coth(x) = 1 / tanh(x) .