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 …

$\text{cosh}\ 2x = \text{cosh}^2x + \text{sinh}^2x= 2 \text{cosh}^2x - 1 = 1 + 2 \text{sinh}^2x$ $\text{tanh}\ 2x = \frac{2\text{tanh}\ x}{1 + \text{tanh}^2x}$ Half 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 …

2024年6月5日 · 在 数学 中， 双曲余弦 是一种 双曲函数，是 双曲几何 中，与欧几里得几何的 余弦函数 相对应的函数。 双曲余弦一般以cosh表示 [1]，在部分较旧的文献中有时会以 表示。 [2] …

The cosh function can calculate on all variables within a table or timetable without indexing to access those variables. All variables must have data types that support the calculation. For …

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 …

2020年9月25日 · 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 …

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 ...

2025年3月5日 · For example, cosz=1/2 (e^ (iz)+e^ (-iz)), (1) so coshz=1/2 (e^z+e^ (-z)). (2) Note that alternate notations are sometimes used, as summarized in the following table. f (x) …

Proof of sinh(x) = cosh(x): From the derivative of e ^x. Given: sinh(x) = ( e ^x - e ^-x)/2; cosh(x) = (e ^x + e ^-x)/2; ( f(x)+g(x) ) = f(x) + g(x); Chain Rule; ( c*f(x) ) = c f(x). Solve: sinh(x)= ( e ^x - e …