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.

最基本的双曲函数是双曲正弦函数 sinh 和双曲余弦函数 cosh，从它们可以导出双曲正切函数 tanh 等，其推导也类似于三角函数的推导。 双曲函数的 反函数 称为 反双曲函数。 双曲函数的 定义域 是实数，其 自变量 的值叫作 双曲角。 双曲函数出现于某些重要的线性微分方程的解中，譬如说定义 悬链线 和 拉普拉斯方程。 返回参数的双曲余弦值。 函数 是关于y轴对称的 偶函数。 函数 是 奇函数。 如同当 遍历实数集时，点 的轨迹是一个圆 一样，当t遍历实数集 时，点 的轨迹是单位 …

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

The area functions are the inverse functions of the hyperbolic functions, i.e., the inverse hyperbolic functions. The functions sinh x, tanh x, and coth x are strictly monotone, so they have unique inverses without any restriction; the function cosh x has two monotonic intervals so we can consider two inverse functions.

2019年1月5日 · According to this walkthrough, argument u u in (cosh(u), sinh(u)) (cosh (u), sinh (u)) should be equal to 2A 2 A, where A A is the area of an intercepted hyperbolic sector from (0, 0) (0, 0) to (cosh(u), sinh(u)) (cosh (u), sinh (u)).

Areasinus hyperbolicus (abgekürzt oder ) und Areakosinus hyperbolicus (abgekürzt oder ) gehören zu den Areafunktionen und sind die Umkehrfunktionen von Sinus hyperbolicus und Kosinus hyperbolicus. Die Funktionen lassen sich durch die folgenden Formeln ausdrücken: Areasinus hyperbolicus: Definition über den natürlichen Logarithmus :

2024年11月25日 · In trigonometry, the coordinates on a unit circle are represented as (cos θ, sin θ), whereas in hyperbolic functions, the pair (cosh θ, sinh θ) represents points on the right half of an equilateral hyperbola. They are used in solving linear differential equations, hyperbolic geometry, and Laplace’s equations in Cartesian coordinates.

2023年10月15日 · We compute the surface area of revolution using ribbons for the hyperbolic cosine function cosh (x) on [-1,1] revolved about the x-axis.

1These are typically read as \kosh of x" and \cinch of x." The parametrization of the hyper-bola H by the area A, resulting in the hyperbolic functions. The parametrization of the unit cir-cle C by the area A, resulting in the circular functions.

Putting θ in radians, this area is α = θ/2 (for instance, the area of the whole circle is π, which is traversed by the full rotation 2π). Looking now at the point intercepted by θ on the circle in terms of α, we have (cos 2α, sin 2α). This is the correct setup for moving to the hyperbolic setting.