📚 Hyperbolic Functions | 双曲函数
The hyperbolic functions sinh, cosh and tanh appear throughout AQA Advanced Level mathematics, especially in calculus, differential equations and further pure topics. They behave like trigonometric functions but are defined through exponential functions, so their graphs, identities and calculus rules follow a distinctive pattern.
双曲函数 sinh、cosh 和 tanh 贯穿 AQA A-Level 数学，尤其是在微积分、微分方程和进阶纯数主题中。它们的性质与三角函数相似，但通过指数函数定义，因此其图像、恒等式和微积分规则具有独特规律。
1. Definitions of sinh, cosh and tanh | 双曲正弦、双曲余弦与双曲正切的定义
The hyperbolic functions are built from the exponential function eˣ. For a real variable x, the two fundamental definitions are sinh x = (eˣ − e⁻ˣ)/2 and cosh x = (eˣ + e⁻ˣ)/2. The hyperbolic tangent is then tanh x = sinh x / cosh x.
双曲函数由指数函数 eˣ 构造。对实变量 x，两个基本定义是 sinh x = (eˣ − e⁻ˣ)/2 和 cosh x = (eˣ + e⁻ˣ)/2。双曲正切则定义为 tanh x = sinh x / cosh x。
sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x/cosh x
e are used in identities and calculus just as sec, cosec and cot are used in trigonometry.
倒数函数为 sech x = 1/cosh x、cosech x = 1/sinh x 和 coth x = 1/tanh x。它们在恒等式和微积分中的使用方式与三角学中的 sec、cosec 和 cot 类似。
The domain of sinh and cosh is all real x. The range of sinh x is ℝ, while the range of cosh x is [1, ∞). The range of tanh x is (−1, 1).
sinh 和 cosh 的定义域都是全体实数。sinh x 的值域为 ℝ，而 cosh x 的值域为 [1, ∞)。tanh x 的值域为 (−1, 1)。
2. Graphs and Key Features | 图像与主要特征
The graph of sinh x is an odd function: it passes through the origin and grows like eˣ/2 for large positive x. The graph of cosh x is even, with a minimum point at (0, 1) and exponential growth in both directions.
sinh x 的图像是奇函数：它经过原点，当 x 取较大正值时像 eˣ/2 一样增长。cosh x 的图像是偶函数，在 (0, 1) 处取最小值，并且向两个方向都呈指数增长。
tal asymptotes at y = 1 and y = −1, and it increases steadily from −1 to 1 as x increases.
tanh x 的图像也是奇函数。它有水平渐近线 y = 1 和 y = −1，并且随着 x 增大从 −1 稳步增加到 1。
| Function | Domain | Range | Symmetry | Key feature |
|---|---|---|---|---|
| sinh x | ℝ | ℝ | odd | passes through origin |
| cosh x | ℝ | [1, ∞) | even | minimum at (0, 1) |
| tanh x | ℝ | (−1, 1) | odd | asymptotes y = ±1 |
3. Osborn's Rule and Fundamental Identities | Osborn 法则与基本恒等式
he hyperbolic analogue of cos² x + sin² x = 1, but the sign between the terms is reversed.
最重要的双曲恒等式是 cosh² x − sinh² x = 1。它是 cos² x + sin² x = 1 的双曲对应式，但两项之间的符号相反。
cosh² x − sinh² x = 1
cosh² x gives 1 − tanh² x = sech² x. Dividing it by sinh² x gives coth² x − 1 = cosech² x.
将该恒等式两边除以 cosh² x 可得 1 − tanh² x = sech² x。两边除以 sinh² x 可得 coth² x − 1 = cosech² x。
1 − tanh² x = sech² x, coth² x − 1 = cosech² x
Osborn' product of two sinh terms. For example, cos² x + sin² x = 1 becomes cosh² x − sinh² x = 1.
Osborn 法则将三角恒等式转换为双曲
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