📚 A-Level AQA Mathematics: Hyperbolic Functions | A-Level AQA 数学：双曲函数 考点精讲
Hyperbolic functions appear in the AQA A-Level Mathematics specification (often within the pure content of Further Mathematics) and combine algebraic manipulation with calculus skills. They are defined via exponential functions, share striking identities with trigonometric functions, and open up a new class of integrals, derivatives and equations. This revision guide covers all key hyperbolic function topics you need to master for your exams.
双曲函数出现在 AQA A-Level 数学考试大纲中（通常属于进阶数学的纯数部分），它将代数运算与微积分技巧紧密结合。双曲函数通过指数函数定义，拥有与三角函数极为相似的恒等式，并引入了全新的积分、求导与方程求解方法。这篇复习指南涵盖了你必须掌握的全部双曲函数核心考点。
1. Introduction to Hyperbolic Functions | 双曲函数简介
Hyperbolic functions are the 'hyperbolic' counterparts of the ordinary trigonometric functions. Instead of being defined on the unit circle, they are related to the unit hyperbola. The two fundamental functions are the hyperbolic sine (sinh x) and hyperbolic cosine (cosh x), from which the hyperbolic tangent (tanh x) and reciprocal functions are derived. They are essential in advanced calculus, differential equations, and engineering applications.
双曲函数是普通三角函数的“双曲”对应版。它们不是基于单位圆定义的，而是与单位双曲线相关。最基本的两个函数是双曲正弦（sinh x）和双曲余弦（cosh x），由它们可以导出双曲正切（tanh x）以及其他倒数函数。双曲函数在高等微积分、微分方程和工程应用中十分重要。
2. Definitions and Exponential Forms | 定义与指数形式
All hyperbolic functions can be expressed cleanly in terms of the exponential function eˣ. The core definitions are:
所有双曲函数都可以用指数函数 eˣ 简洁表达。核心定义如下：
sinh x = (eˣ − e⁻ˣ) / 2
cosh x = (eˣ + e⁻ˣ) / 2
tanh x = sinh x / cosh x = (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ)
exponential forms are the key to proving identities, solving equations and differentiating.
由此还可以定义倒数函数：sech x = 1/cosh x，cosech x = 1/sinh x，coth x = 1/tanh x。这些指数形式是证明恒等式、解方程和求导的基础。
3. Graphs of sinh x, cosh x, tanh x | sinh x, cosh x, tanh x 的图像
The graph of y = sinh x is an odd function passing through (0,0). It resembles a cubic curve but grows like ±½ eˣ for large |x|. y = cosh x ven function with minimum value 1 at x = 0; it is the shape of a hanging chain (catenary). y = tanh x sigmoid function with horizontal asymptotes y = 1 and y = −1, passing through the origin.
y = sinh x 的图像是奇函数，穿过原点，形状类似三次曲线，但在大 |x| 时以 ±½ eˣ 的速度增长。y = cosh x 是偶函数，在 x=0 处取得最小值1，形状像悬挂的链条（悬链线）。y = tanh x 是奇函数，呈 S 形，有两条水平渐近线 y=1 和 y=−1，且过原点。
tching: cosh x ≥ 1, tanh x is bounded between −1 and 1, and both sinh x and tanh x are one‑to‑one, so they have inverses on their entire domains. cosh x is not one‑to‑one, so its inverse requires a restricted domain x ≥ 0.
作图关键点：cosh x ≥ 1，tanh x 被限制在 −1 到 1 之间；sinh x 和 tanh x 是一一对应的，因此在整个定义域上都存在反函数。而 cosh x 不是一一对应的，所以它的反函数需要限制定义域为 x ≥ 0。
4. Hyperbolic Identities | 双曲恒等式
Hyperbolic identities closely mirror trigonometric ones, but with sign changes. The most fundamental is the hyperbolic Pythagorean identity:
双曲恒等式与三角恒等式非常相似，但符号有所不同。最基本的恒等式是双曲毕达哥拉斯恒等式：
cosh²x − sinh²x = 1
From this, dividing by cosh²x or sinh²x gives:
在此基础上，分别除以 cosh²x 或 sinh²x 可得：
1 − tanh²x = sech²x
coth²x − 1 = cosech²x
Double‑angle formulas are analogous to trigonometry:
倍角公式与三角中的情形类似：
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh²x + sinh²x = 2 cosh²x − 1 = 1 + 2 sinh²x
sions, which follow the same structure as for sine and cosine but with attention to signs.
其他恒等式还包括 sinh(x ± y) 和 cosh(x ± y) 的展开式，它们在结构上与正弦和余弦的展开式相同，但需要注意符号变化。
5. Inverse Hyperbolic Functions | 反双曲函数
The inverse hyperbolic functions are denoted arsinh, arcosh, artanh (or sometimes arcsinh etc. in older notation). Their domains and ranges are crucial:
反双曲函数通常记作 arsinh, arcosh, artanh（旧标准有时也写作 arcsinh 等）。它们的定义域和值域非常重要：
| Function | Domain | Range |
|---|---|---|
| y = arsinh x | ℝ | ℝ |
| y = arcosh x | x ≥ 1 | y ≥ 0 |
| y = artanh x | |x| < 1 | ℝ |
explicitly, and they can be expressed in logarithmic form, which is a favourite exam task.
这些反函数使我们能够显式地解出形如 sinh y = x 的方程，而且它们都可以用对数形式表达，这是考试中常见的题型。
6. Logarithmic Forms of Inverse Functions | 反函数的对数形式
Using the exponential definitions, we can derive exact logarithmic expressions for the inverse hyperbolic functions:
利用指数定义，我们可以推导出反双曲函数的精确对数表达式：
arsinh x = ln(x + √x² + 1)
arcosh x = ln(x + √x² − 1), x ≥ 1
artanh x = ½ ln((1 + x)/(1 − x)), |x| < 1
These formulas are especially useful when integrating functions that lead to inverse hyperbolic forms, or when solving equations where you need an exact simplified answer.
这些公式在积分得到反双曲形式时尤其有用，也常用于解方程并给出精确的化简答案。
7. Differentiation of Hyperbolic Functions | 双曲函数的微分
The derivatives of hyperbolic and inverse hyperbolic functions must be memorised. They are all standard results:
双曲函数与反双曲函数的导数必须熟记。它们都是标准结论：
- d/dx (sinh x) = cosh x | 双曲正弦的导数
- d/dx (cosh x) = sinh x | 双曲余弦的导数
- d/dx (tanh x) = sech²x | 双曲正切的导数
- d/dx (sech x) = −sech x tanh x | 双曲正割的导数
- d/dx (cosech x) = −cosech x coth x | 双曲余割的导数
- d/dx (coth x) = −cosech²x | 双曲余切的导数
For inverse functions:
对于反双曲函数：
- d/dx (arsinh x) = 1 / √x² + 1 | 反双曲正弦的导数
- d/dx (arcosh x) = 1 / √x² − 1 | 反双曲余弦的导数
- d/dx (artanh x) = 1 / (1 − x²) | 反双曲正切的导数
Notice the similarity to the derivatives of inverse trigonometric functions, but without the sign changes in the radicand. The chain rule then extends these to differentiate composite functions such as sinh(ax + b) or artanh(x²).
请注意其与反三角函数导数的相似性，但被开方数的符号没有变化。结合链式法则，就可以对复合函数求导，比如 sinh(ax + b) 或 artanh(x²)。
8. Integration of Hyperbolic Functions | 双曲函数的积分
Integration reverses differentiation, so the basic integrals are immediate:
积分是微分的逆运算，因此基本积分公式可以直接得出：
∫ sinh x dx = cosh x + C
∫ cosh x dx = sinh x + C
∫ tanh x dx = ln cosh x + C
Moreover, the derivatives of inverse functions lead to important standard integrals that often appear on the AQA formula sheet or must be known:
此外，由反函数的导数可以引出一系列重要的标准积分，这些积分常出现在 AQA 公式表中或需要考生掌握：
∫ 1/√x² + a² dx = arsinh(x/a) + C = ln|x + √x² + a²| + C
∫ 1/√x² − a² dx = arcosh(x/a) + C = ln|x + √x² − a²| + C
∫ 1/(a² − x²) dx = (1/a) artanh(x/a) + C, |x| < a
These integrals can also be evaluated using trigonometric substitutions, but hyperbolic forms often provide a more elegant solution.
这些积分也可以用三角换元法计算，但双曲形式往往能给出更简洁优美的结果。
9. Solving Equations with Hyperbolic Functions | 解双曲函数方程
ly involves converting to exponentials or using identities. For example, solve sinh x = 3:
考试中经常要求解含有双曲函数的方程。常用的策略是转化为指数形式或运用恒等式。例如解 sinh x = 3：
Rewrite as (eˣ − e⁻ˣ)/2 = 3 ⇒ eˣ − e⁻ˣ = 6 ⇒ multiply by eˣ: e²ˣ − 1 = 6eˣ ⇒ e²ˣ − 6eˣ − 1 = 0, a quadratic in eˣ. Solving gives eˣ = 3 + √10 (positive root). Hence x = ln(3 + √10). Alternatively, directly use the inverse: x = arsinh 3 = ln(3 + √10).
写成 (eˣ − e⁻ˣ)/2 = 3 ⇒ eˣ − e⁻ˣ = 6 ⇒ 两边乘 eˣ：e²ˣ − 1 = 6eˣ ⇒ e²ˣ − 6eˣ − 1 = 0，这是关于 eˣ 的二次方程。解得 eˣ = 3 + √10（取正根），因此 x = ln(3 + √10)。或者直接使用反函数：x = arsinh 3 = ln(3 + √10)。
2x = 5, are handled by expressing everything in exponentials. Since cosh 2x + sinh 2x = e²ˣ, the equation simplifies to e²ˣ lways check for extraneous solutions when squaring or using identities that alter domains.
对于更复杂的方程，例如 cosh 2x + sinh 2x = 5，可以把所有函数写成指数形式。因为 cosh 2x + sinh 2x = e²ˣ，方程简化为 e²ˣ = 5，因此 2x = ln 5，x = (1/2) ln 5。在平方或使用恒等式改变定义域时，一定要检验是否有增根。
10. Exam Tips and Summary | 考试技巧与总结
al definitions: they unlock proofs, derivatives and solutions. Memorise the key identities – especially cosh²x − sinh²x = 1 and the double‑angle forms – always include the constant of integration. In equation solving, converting to exponentials is often the safest route, but inverse functions and their logarithmic forms can save time. Finally, practise sketching graphs and stating domains and ranges; these are common low‑tariff marks that are easily won.
要精通双曲函数，就要牢记指数定义：它们是证明、求导和解方程的基础。熟记关键恒等式——尤其是 cosh²x − sinh²x = 1 以及倍角公式——它们对化简表达式至关重要。在积分时，识别何时可以用反双曲函数表达结果，并永远不忘加积分常数。解方程时，转化为指数形式通常是最稳妥的路径，但反函数及其对数形式也能节省大量时间。最后，多练习图像绘制并准确陈述定义域和值域；这些是常见的低分值题目，容易得分。
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