📚 Hyperbolic Functions Revision for CCEA A-Level Mathematics | A-Level CCEA 数学：双曲函数 考点精讲
Hyperbolic functions appear throughout the CCEA A-Level Mathematics and Further Mathematics specifications, often catching students off guard because they combine exponentials, identities, graphs and calculus in ways that resemble trigonometry yet behave differently. This revision guide covers every essential topic – from definitions and identities to derivatives, integrals and solving equations – ensuring you can approach exam questions with confidence.
双曲函数贯穿 CCEA A-Level 数学和高阶数学的考纲，常常让考生措手不及，因为它们以指数函数为基础，融合了恒等式、图像和微积分，形式上类似于三角函数，但性质不同。本文梳理所有必备考点——从定义和恒等式，到导数、积分和解方程——帮助你自信应对考试。
1. Definitions of Hyperbolic Functions | 双曲函数的定义
The two fundamental hyperbolic functions are defined in terms of exponential functions: sinh x = (ex – e-x)/2 and cosh x = (ex + e-x)/2. The remaining four functions are derived from these: tanh x = sinh x / cosh x, coth x = 1 / tanh x (cosh x / sinh x), sech x = 1 / cosh x, and csch x (or cosech x) = 1 / sinh x.
两个基本的双曲函数用指数函数定义：sinh x = (ex – e-x)/2 和 cosh x = (ex + e-x)/2。其余四个函数均由它们派生：tanh x = sinh x / cosh x，coth x = 1 / tanh x（即 cosh x / sinh x），sech x = 1 / cosh x，以及 csch x（或 cosech x）= 1 / sinh x。
Note that sinh x is an odd function, cosh x is an even function, and tanh x is odd. Their domain is all real numbers, while the ranges differ: sinh x has range ℝ; cosh x has range [1, ∞); tanh x has range (-1, 1).
注意 sinh x 是奇函数，cosh x 是偶函数，tanh x 是奇函数。它们的定义域都是全体实数，但值域不同：sinh x 的值域为 ℝ；cosh x 的值域为 [1, ∞)；tanh x 的值域为 (-1, 1)。
2. Fundamental Identities | 基本恒等式
The hyperbolic equivalent of the Pythagorean identity is cosh²x – sinh²x = 1. Dividing through by cosh²x gives 1 – tanh²x = sech²x, and dividing by sinh²x gives coth²x – 1 = csch²x.
双曲函数中的“毕达哥拉斯恒等式”是 cosh²x – sinh²x = 1。两边除以 cosh²x 得 1 – tanh²x = sech²x；除以 sinh²x 得 coth²x – 1 = csch²x。
Other useful identities include the double-argument formulas: sinh 2x = 2 sinh x cosh x, and cosh 2x = cosh²x + sinh²x = 2 cosh²x – 1 = 1 + 2 sinh²x. These are direct consequences of the definitions and mirror their trigonometric counterparts with sign changes governed by Osborn’s rule.
其他有用的恒等式包括倍角公式：sinh 2x = 2 sinh x cosh x，以及 cosh 2x = cosh²x + sinh²x = 2 cosh²x – 1 = 1 + 2 sinh²x。这些都可以由定义直接导出，在形式上与三角恒等式相似，但符号变化遵循 Osborn 法则。
3. Graphs and Properties | 图形与性质
The graph of y = sinh x passes through the origin and is strictly increasing, resembling a cubic curve but growing exponentially for large |x|. y = cosh x is a symmetric curve with minimum at (0, 1), often called the catenary. y = tanh x has horizontal asymptotes at y = ±1 and passes through the origin with an S-shaped profile.
y = sinh x 的图像过原点且严格单调递增，外形类似三次曲线，但在 |x| 很大时呈指数增长。y = cosh x 是对称曲线，最低点为 (0, 1)，常被称为悬链线。y = tanh x 有水平渐近线 y = ±1，过原点，呈 S 形。
You should be able to sketch these graphs and identify key features: intercepts, asymptotes, symmetry and monotonic intervals. These sketches are vital when solving inequalities or understanding inverse functions.
你必须能够画出这些图像并标注关键特征：截距、渐近线、对称性和单调区间。这些草图在解不等式或理解反函数时至关重要。
4. Inverse Hyperbolic Functions | 反双曲函数
The inverse hyperbolic functions are denoted arsinh x, arcosh x and artanh x. Their domains and principal branches are: arsinh x has domain ℝ; arcosh x has domain [1, ∞) and range [0, ∞); artanh x has domain (-1, 1) and range ℝ. The derivatives of these inverse functions will be covered later.
反双曲函数记作 arsinh x、arcosh x 和 artanh x。它们的定义域与主值分支为：arsinh x 的定义域是 ℝ；arcosh x 的定义域是 [1, ∞)，值域是 [0, ∞)；artanh x 的定义域是 (-1, 1)，值域是 ℝ。这些反函数的导数将在后面讨论。
When solving equations such as sinh x = k, we write x = arsinh k. Your calculator may use the notation sinh⁻¹, but the A-level specification expects fluency with both name forms.
解方程 sinh x = k 时，可写为 x = arsinh k。你的计算器上可能会显示 sinh⁻¹，但 A-level 考纲要求能熟练使用这两种记法。
5. Logarithmic Forms | 对数形式
The inverse hyperbolic functions can be expressed using natural logarithms. The standard logarithmic forms are: arsinh x = ln(x + √(x² + 1)) for all real x, arcosh x = ln(x + √(x² – 1)) for x ≥ 1, and artanh x = ½ ln((1 + x)/(1 – x)) for |x| < 1.
反双曲函数可以用自然对数表示。标准对数形式为：arsinh x = ln(x + √(x² + 1)) 对所有实数 x成立，arcosh x = ln(x + √(x² – 1)) 对 x ≥ 1 成立，以及 artanh x = ½ ln((1 + x)/(1 – x)) 对 |x| < 1 成立。
These logarithmic forms are extremely useful for exact evaluation and for solving equations where the argument is a simple fraction. For instance, artanh(½) = ½ ln 3.
这些对数形式在精确计算以及求解自变量为简单分数的方程时极其有用。例如，artanh(½) = ½ ln 3。
6. Derivatives | 导数
The derivatives of the basic hyperbolic functions are straightforward: d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech²x. The derivatives of coth x, sech x and csch x follow from standard rules: d/dx (coth x) = -csch²x, d/dx (sech x) = -sech x tanh x, d/dx (csch x) = -csch x coth x.
基本双曲函数的导数非常直接：d/dx (sinh x) = cosh x，d/dx (cosh x) = sinh x，d/dx (tanh x) = sech²x。coth x、sech x 和 csch x 的导数可由链式法则得到：d/dx (coth x) = -csch²x，d/dx (sech x) = -sech x tanh x，d/dx (csch x) = -csch x coth x。
For the inverse functions, the derivatives are: d/dx (arsinh x) = 1/√(x² + 1), d/dx (arcosh x) = 1/√(x² – 1) (x > 1), and d/dx (artanh x) = 1/(1 – x²) for |x| < 1. These can be derived via implicit differentiation or using the logarithmic forms.
反双曲函数的导数为：d/dx (arsinh x) = 1/√(x² + 1)，d/dx (arcosh x) = 1/√(x² – 1) (x > 1)，以及 d/dx (artanh x) = 1/(1 – x²) 对 |x| < 1 成立。它们可以通过隐函数求导或由对数形式推导得到。
7. Integrals | 积分
Integration of hyperbolic functions is the reverse of differentiation. The standard integrals are: ∫ sinh x dx = cosh x + C, ∫ cosh x dx = sinh x + C, ∫ tanh x dx = ln(cosh x) + C, and ∫ sech²x dx = tanh x + C.
双曲函数的积分即为求导的逆运算。标准积分包括：∫ sinh x dx = cosh x + C，∫ cosh x dx = sinh x + C，∫ tanh x dx = ln(cosh x) + C，以及 ∫ sech²x dx = tanh x + C。
For more complicated integrals, recognizing the form ∫ f'(x)/√(f(x)² ± a²) dx or using substitution often leads to inverse hyperbolic functions. The integral ∫ 1/√(x² + a²) dx evaluates to arsinh(x/a) + C, while ∫ 1/√(x² – a²) dx (with x > a) gives arcosh(x/a) + C, and ∫ 1/(a² – x²) dx gives (1/a) artanh(x/a) + C for |x| < a.
对于更复杂的积分，识别形如 ∫ f'(x)/√(f(x)² ± a²) dx 的形式，或采用换元法，往往能得到反双曲函数。积分 ∫ 1/√(x² + a²) dx 结果为 arsinh(x/a) + C，∫ 1/√(x² – a²) dx (x > a) 为 arcosh(x/a) + C，而 ∫ 1/(a² – x²) dx 当 |x| < a 时给出 (1/a) artanh(x/a) + C。
8. Solving Equations Involving Hyperbolics | 双曲方程的求解
CCEA exam questions frequently ask you to solve equations such as sinh x = 2 or cosh 2x + 3 sinh x = 1. For simple cases, use the logarithmic forms directly. For more involved equations, apply the exponential definitions or hyperbolic identities to reduce the equation to a quadratic in ex or in a single hyperbolic function.
CCEA 考题经常要求解如 sinh x = 2 或 cosh 2x + 3 sinh x = 1 的方程。对于简单情形，可直接使用对数形式。对于更复杂的方程，应用指数定义或双曲恒等式，将方程化为关于 ex 或单个双曲函数的二次方程来求解。
Example: Solve 3 sinh x – 4 cosh x = 2. Write sinh x and cosh x in terms of ex and e-x, multiply through by ex, and obtain a quadratic in ex. Always check for extraneous solutions when squaring or using logarithmic forms.
例：求解 3 sinh x – 4 cosh x = 2。将 sinh x 和 cosh x 用 ex 和 e-x 表示，两端乘以 ex，得到一个关于 ex 的二次方程。在平方或使用对数形式时，务必验根以排除增根。
9. Osborn’s Rule and Connections to Trigonometry | Osborn 法则及其与三角的联系
Osborn’s rule states that any trigonometric identity can be converted to the corresponding hyperbolic identity by replacing each trigonometric function with its hyperbolic counterpart, and changing the sign of any term containing a product of two sines. For example, from sin²x + cos²x = 1 we get cosh²x – sinh²x = 1 (the sign changes because of the product of two sines, sin²x counts as a product).
Osborn 法则指出：任何一个三角恒等式都可以转化为相应的双曲恒等式，只需将每个三角函数替换为对应的双曲函数，并将任何包含两个正弦乘积的项的符号改变。例如，从 sin²x + cos²x = 1 可得到 cosh²x – sinh²x = 1（符号改变是因为存在两个正弦的乘积，sin²x 视为一个乘积）。
This rule explains why the derivatives and identities differ in sign patterns. It is a powerful mnemonic for checking your work and understanding the parallels between circular and hyperbolic functions.
这条法则解释了为什么导数和恒等式中符号模式有所不同。它是一个强大的记忆工具，用于检查你的解答，并理解圆函数与双曲函数之间的对应关系。
10. Applications: Catenary | 应用：悬链线
A classic application is the catenary – the shape of a hanging flexible chain or cable under uniform gravity. Its equation is y = a cosh(x/a), where a is a constant related to the tension and weight per unit length. The lowest point is at x = 0, and the shape is symmetric.
一个经典应用是悬链线——在均匀重力作用下悬挂的柔软链条或电缆的形状。其方程为 y = a cosh(x/a)，其中 a 是一个与张力和单位长度重量相关的常数。最低点在 x = 0 处，形状对称。
You may be asked to find the length of a catenary segment or the gradient at a point using hyperbolic derivatives. The arc length from 0 to x is s = a sinh(x/a), which involves integrating √(1 + (dy/dx)²). This demonstrates the practical importance of hyperbolic calculus.
考试中可能会要求用双曲导数求悬链线段的长度或
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导