\displaystyle \text {cosh}\ x - \text {cosh}\ y = 2 \text {sinh}\ \frac12 (x + y)\ \text {sinh}\ \frac12 (x - y) cosh x−cosh y = 2sinh 21(x+y) sinh 21(x−y)

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

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.

函数 cosh x 是关于 y 轴对称的偶函数。 函数 sinh x 是奇函数，就是说 -sinh x = sinh (-x) 且 sinh 0 = 0。 y=sinh x，定义域：R，值域：R，奇函数，函数图像为过原点并且穿越Ⅰ、Ⅲ象限的严格单调递增曲线，函数图像关于原点对称。 [1] y=cosh x，定义域：R，值域： [1,+∞)，偶函数，函数图像是 悬链线，最低点是（0，1），在Ⅰ象限部分是严格单调递增曲线，函数图像关于y轴对称。 y=tanh x，定义域：R，值域： (-1,1)，奇函数，函数图像为过原点并且穿越Ⅰ、Ⅲ象限的严格 …

在 数学 中， 双曲函数 是一类与常见的 三角函数 （也叫圆函数）类似的函数。 最基本的双曲函数是 雙曲正弦 函数 和 雙曲餘弦 函数 ，从它们可以导出 双曲正切 函数 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，其自变量的值叫做 双曲角。 双曲函数出现于某些重要的线性 微分方程 的解中，譬如說定义 悬链线 和 拉普拉斯方程。 最簡單的幾種雙曲函數為 [1]： {\displaystyle \tanh x= {\frac {\sinh x} {\cosh x}}= {\frac {e^ {x} …

In Mathematics, the hyperbolic functions are similar to the trigonometric functions or circular functions. Generally, the hyperbolic functions are defined through the algebraic expressions that include the exponential function (e x) and its inverse exponential functions (e -x), where e is the Euler’s constant.

Math2.org Math Tables: Hyperbolic Trigonometric Identities(Math)

\color {blue} {\textbf {双曲函数的定义}} 1、 双曲正弦. \sinh x = \frac {e^ {x} - e^ {-x}} {2} \\ 2、 双曲余弦. \cosh x = \frac {e^ {x} + e^ {-x}} {2} \\ 3、 双曲正切. \sinh x =\frac {\sinh} {\cosh} = \frac {e^ {x} - e^ {-x}} {e^ {x} + e^ {-x}} \\ 4、 双曲余切. \coth x = \frac {\cosh x} {\sinh x} =\frac {e^ {x} + e^ {-x}} {e^ {x} - e^ {-x}} \\ 5、 双曲正割.

Hyperbolic functions formulas - Sinh x, Cosh x, Tanh x & more. Properties of hyperbolic functions, Sample Problems on Hyperbolic functions, examples & more.

Hyperbolic Trig Identities is like trigonometric identities yet may contrast to it in specific terms. The fundamental hyperbolic functions are hyperbola sin and hyperbola cosine from which the other trigonometric functions are inferred. You can easily explore …