\sinh \frac{x}{2}=\pm \sqrt{\frac{\cosh x-1}{2}}\left(\begin{array}{l} x>0, \qquad \text{取正号} \\ x<0,\qquad \text{取负号} \end{array} \right)\\ \cosh \frac{x}{2}=\sqrt{\frac{\cosh x+1}{2}} \\ \tanh \frac{x}{2}=\frac{\cosh x-1}{\sinh x}=\frac{\sinh x}{\cosh x+1} \\ \coth \frac{x}{2}=\frac{\sinh x}{\cosh x-1}=\frac{\cosh x+1}{\sinh x} \\

cosh 2 (x) - sinh 2 (x) = 1 tanh 2 (x) + sech 2 (x) = 1 coth 2 (x) - csch 2 (x) = 1 Inverse Hyperbolic Defintions. arcsinh(z) = ln( z + (z 2 + 1) ) arccosh(z) = ln( z (z 2 - 1) ) arctanh(z) = 1/2 ln( (1+z)/(1-z) ) arccsch(z) = ln( (1+ (1+z 2) )/z ) arcsech(z) = ln( (1 (1-z 2) )/z ) arccoth(z) = 1/2 ln( (z+1)/(z-1) ) Relations to Trigonometric ...

Osborn's rule指出：将圆三角函数恒等式中，圆函数转成相应的双曲函数，有两个 \sinh 的积时，包括 \coth ^2\left ( x \right) ,\tanh ^2\left ( x \right) ,\mathrm {csch} ^2\left ( x \right) ,\sinh \left ( x \right) \cdot \sinh \left ( y \right) ，则转换正负号，则可得到相应的双曲函数恒等式。

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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解释一下就是：我们先得到一个三角恒等式，然后把里面的sin或者cos全部转为sinh或cosh，如果在公式中出现两个sin相乘，那么把前面的符号改成负号。 上面也给了一个例子，是关于cos 2A的二倍角公式的，大家可以很清楚的看到就是根据Osborn's Rule来进行转化的 ...

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

2024年10月28日 · 双曲正弦（sinh）和双曲余弦（cosh）是双曲函数中最基本的两个函数，它们的定义如下： - 双曲正弦函数（sinh）定义为： \[ \sinh(x) = \frac{e^x - e^{-x}}{2} \] - 双曲余弦函数（cosh）定义为：

2023年7月28日 · One of the fundamental hyperbolic trig identities is the hyperbolic Pythagorean identity: cosh^2(x) - sinh^2(x) = 1. This identity bears resemblance to the Pythagorean identity in circular trigonometry, but here we deal with hyperbolic functions.

The hyperbolic functions satisfy a number of identities. These allow expressions involving the hyperbolic functions to be written in different, yet equivalent forms. Several commonly used identities are given on this leaflet. 1. Hyperbolic identities.