📚 Hyperbolic Functions for IGCSE CCEA Maths: Key Revision Notes | IGCSE CCEA 数学：双曲函数 考点精讲
Hyperbolic functions are an important family of functions defined using exponential functions. They share many algebraic properties with trigonometric functions but have distinct graphs and uses. In this article, we will explore the definitions, graphs, identities, inverse functions, and basic solving techniques for sinh x, cosh x, and tanh x, covering the key points you need for IGCSE CCEA Mathematics.
双曲函数是一类用指数函数定义的重要函数。它们与三角函数有许多相似的代数性质，但具有不同的图像和用途。本文将探索 sinh x、cosh x 和 tanh x 的定义、图像、恒等式、反函数和基本求解技巧，涵盖 IGCSE CCEA 数学所需掌握的关键考点。
1. What Are Hyperbolic Functions? | 什么是双曲函数？
, written as sinh x, cosh x, tanh x, etc., are combinations of the exponential functions eˣ and e⁻ˣ. They are called 'hyperbolic' because the point (cosh t, sinh t) lies on the right branch of the hyperbola x² − y² = 1.
双曲函数，记作 sinh x、cosh x、tanh x 等，是由指数函数 eˣ 和 e⁻ˣ 组合而成的。之所以称为“双曲”函数，是因为点 (cosh t, sinh t) 位于双曲线 x² − y² = 1 的右支上。
The basic hyperbolic functions are defined for all real numbers. They appear in many areas such as calculus, physics, and engineering. For IGCSE CCEA Maths, you need to recognise their definitions and properties.
基本双曲函数对所有实数有定义。它们出现在微积分、物理和工程等许多领域。对于 IGCSE CCEA 数学，你需要认识它们的定义和性质。
2. Definition of sinh x | sinh x 的定义
The hyperbolic sine function is defined as:
双曲正弦定义为：
sinh x = (eˣ − e⁻ˣ) / 2
g sinh(−x) = −sinh x. It has a domain of all real numbers and range also all real numbers.
sinh x 是奇函数，意味着 sinh(−x) = −sinh x。其定义域为全体实数，值域也为全体实数。
ponential expression. For example, sinh 0 = (1 − 1)/2 = 0. sinh(ln 2) = (2 − 1/2)/2 = 3/4.
要计算给定 x 的 sinh x，只需将值代入指数表达式。例如，sinh 0 = (1 − 1)/2 = 0。sinh(ln 2) = (2 − 1/2)/2 = 3/4。
3. Definition of cosh x | cosh x 的定义
The hyperbolic cosine function is defined as:
双曲余弦定义为：
cosh x = (eˣ + e⁻ˣ) / 2
al numbers, and its range is [1, ∞). The minimum value of cosh x is 1, occurring at x = 0.
cosh x 是偶函数：cosh(−x) = cosh x。其定义域为全体实数，值域为 [1, ∞)。cosh x 的最小值为 1，出现在 x = 0 处。
This function describes the shape of a hanging cable (catenary). In IGCSE questions, you might be asked to evaluate cosh x exactly or to simplify expressions involving cosh x.
这个函数描述了悬链线的形状。在 IGCSE 考题中，可能会让你准确计算 cosh x 或化简含 cosh x 的表达式。
4. Definition of tanh x | tanh x 的定义
The hyperbolic tangent is defined as the ratio of sinh x to cosh x:
双曲正切定义为 sinh x 与 cosh x 的比值：
tanh x = sinh x / cosh x = (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ)
ntal asymptotes y = 1 and y = −1. Its domain is all real numbers and its range is (−1, 1).
tanh x 是奇函数，有水平渐近线 y = 1 和 y = −1。其定义域为全体实数，值域为 (−1, 1)。
s often used to model saturation processes. Understanding its behaviour helps in graph sketching.
由于当 x → ±∞ 时 tanh x 趋向 ±1，它常用于建模饱和过程。了解其行为有助于图像绘制。
5. Graphs of Hyperbolic Functions | 双曲函数图像
Recognising the graphs of sinh x, cosh x, and tanh x is a common exam requirement. Here are their key features:
识别 sinh x、cosh x 和 tanh x 的图像是常见的考试要求。以下是它们的主要特征：
- y = sinh x: Passes through origin, odd, monotonic increasing, no asymptotes.
- y = cosh x: Even, minimum at (0,1), U-shaped, symmetric about y-axis, no asymptotes.
- y = tanh x: Odd, passes through origin, S-shaped curve, bounded by horizontal asymptotes y = ±1.
- y = sinh x：经过原点，奇函数，单调递增，无渐近线。
- y = cosh x：偶函数，在 (0,1) 处取最小值，U 形，关于 y 轴对称，无渐近线。
- y = tanh x：奇函数，经过原点，S 形曲线，由水平渐近线 y = ±1 界定。
Use graphing software or a table of values to plot these functions. In an exam, you may need to match equations to given graphs or sketch transformations.
使用绘图软件或数值表绘制这些函数。在考试中，你可能需要将方程与给定图像匹配，或绘制变换后的图像。
6. Fundamental Hyperbolic Identity | 基本双曲恒等式
The most important identity linking cosh x and sinh x is:
连接 cosh x 和 sinh x 的最重要的恒等式是：
cosh² x − sinh² x = 1
1 in trigonometry, but with a minus sign. It can be proved directly from the definitions.
这类似于三角恒等式 cos² θ + sin² θ = 1，但这里是减号。可以直接从定义证明。
e sech x = 1 / cosh x. These are useful for simplifying expressions and solving equations.
由此可推出其他恒等式，如 1 − tanh² x = sech² x，其中 sech x = 1 / cosh x。这些对化简表达式和解方程很有用。
7. Osborn's Rule and Related Identities | 奥斯本法则及相关恒等式
Many trigonometric identities can be converted into hyperbolic identities using Osborn's Rule: replace the trigonometric function with its hyperbolic counterpart, and change the sign of any product (or implied product) of two sine terms.
许多三角恒等式可以使用奥斯本法则转换为双曲恒等式：将三角函数替换为对应的双曲函数，并改变任何两个正弦项乘积（或隐含乘积）的符号。
For example, the double-angle formulas:
例如，倍角公式：
- sin 2x = 2 sin x cos x → sinh 2x = 2 sinh x cosh x
- cos 2x = cos² x − sin² x → cosh 2x = cosh² x + sinh² x
n the cosh 2x identity: the minus becomes a plus. Similarly, cos 2x = 2 cos² x − 1 becomes cosh 2x = 2 cosh² x − 1, and cos 2x = 1 − 2 sin² x becomes cosh 2x = 1 + 2 sinh² x.
注意 cosh 2x 恒等式中符号的变化：减号变成加号。类似地，cos 2x = 2 cos² x − 1 变成 cosh 2x = 2 cosh² x − 1，而 cos 2x = 1 − 2 sin² x 变成 cosh 2x = 1 + 2 sinh² x。
8. Inverse Hyperbolic Functions | 反双曲函数
The inverse hyperbolic functions are denoted arsinh x, arcosh x, artanh x (or sinh⁻¹ x, etc.). They can be expressed in terms of natural logarithms.
反双曲函数记作 arsinh x、arcosh x、artanh x（或 sinh⁻¹ x 等）。它们可用自然对数表示。
- arsinh x = ln(x + √x² + 1), domain: all real x
- arcosh x = ln(x + √x² − 1), domain: x ≥ 1
- artanh x = ½ ln((1 + x) / (1 − x)), domain: |x| < 1
- arsinh x = ln(x + √x² + 1)，定义域：所有实数 x
- arcosh x = ln(x + √x² − 1)，定义域：x ≥ 1
- artanh x = ½ ln((1 + x) / (1 − x))，定义域：|x| < 1
seful for solving equations like sinh x = a, where the exact solution is arsinh a = ln(a + √a² + 1).
这些对数形式对于解像 sinh x = a 这样的方程尤其有用，其精确解为 arsinh a = ln(a + √a² + 1)。
9. Solving Hyperbolic Equations | 解双曲方程
s, often you convert everything to exponentials or use the identity cosh² x − sinh² x = 1.
要解涉及双曲函数的方程，通常将一切化为指数形式，或使用恒等式 cosh² x − sinh² x = 1。
Example: Solve 7 sinh x + 3 cosh x = 4.
例：解方程 7 sinh x + 3 cosh x = 4。
Write sinh x and cosh x in exponential form: 7((eˣ − e⁻ˣ)/2) + 3((eˣ + e⁻ˣ)/2) = 4. Multiply by 2: 7eˣ − 7e⁻ˣ + 3eˣ + 3e⁻ˣ = 8 → 10eˣ − 4e⁻ˣ = 8.
将 sinh x 和 cosh x 写为指数形式：7((eˣ − e⁻ˣ)/2) + 3((eˣ + e⁻ˣ)/2) = 4。两边乘以 2：7eˣ − 7e⁻ˣ + 3eˣ + 3e⁻ˣ = 8 → 10eˣ − 4e⁻ˣ = 8。
Multiply by eˣ: 10e²ˣ − 4 = 8eˣ. Rearrange: 10e²ˣ − 8eˣ − 4 = 0. Divide by 2: 5e²ˣ − 4eˣ − 2 = 0. Let y = eˣ, then 5y² − 4y − 2 = 0. Solve for y, discard negative root, and finally x = ln y.
乘以 eˣ：10e²ˣ − 4 = 8eˣ。整理：10e²ˣ − 8eˣ − 4 = 0。除以 2：5e²ˣ − 4eˣ − 2 = 0。令 y = eˣ，则 5y² − 4y − 2 = 0。解出 y，舍去负根，最后 x = ln y。
This method works for linear combinations of sinh and cosh. Sometimes using the identity first simplifies the work.
这种方法适用于 sinh 和 cosh 的线性组合。有时先使用恒等式可以简化过程。
10. Derivatives of Hyperbolic Functions | 双曲函数的导数
Although differentiation is part of IGCSE CCEA further concepts, knowing the derivatives of hyperbolic functions can be helpful for extension problems. The rules are straightforward:
虽然微分是 IGCSE CCEA 拓展概念的一部分，但了解双曲函数的导数有助于解决拓展问题。规则很简单：
- d/dx (sinh x) = cosh x
- d/dx (cosh x) = sinh x
- d/dx (tanh x) = sech² x
ithout the sign changes. For instance, d/dx (cos x) = −sin x, but d/dx (cosh x) = +sinh x.
注意与三角函数导数的对称性，但没有符号变化。例如，d/dx (cos x) = −sin x，但 d/dx (cosh x) = +sinh x。
This often catches students out. Always check whether you are dealing with circular or hyperbolic functions.
这常常让学生犯错。一定要检查你处理的是三角函数还是双曲函数。
11. Key Identities Summary Table | 关键恒等式汇总表
Here is a quick reference table for hyperbolic identities you are expected to know:
下面是你应该知道的恒等式快速参考表：
| cosh² x − sinh² x = 1 | 1 − tanh² x = sech² x |
| sinh(2x) = 2 sinh x cosh x | cosh(2x) = cosh² x + sinh² x |
| cosh(2x) = 2 cosh² x − 1 | cosh(2x) = 1 + 2 sinh² x |
| sinh(x + y) = sinh x cosh y + cosh x sinh y | cosh(x + y) = cosh x cosh y + sinh x sinh y |
These identities, together with the definitions, will allow you to tackle most hyperbolic problems in the IGCSE CCEA syllabus.
这些恒等式连同定义，将使你能应对 IGCSE CCEA 大纲中的大多数双曲函数问题。
12. Tips for the Exam | 考试技巧
ch a quick graph to visualise the range or asymptotes. Check your solutions using the identity cosh² x − sinh² x = 1.
面对双曲函数问题时，始终先写出相关定义。如果卡住了，就转化为指数形式。快速绘制草图以直观了解值域或渐近线。用恒等式 cosh² x − sinh² x = 1 检验解。
Be careful with signs when applying Osborn's Rule. Practice differentiating between hyperbolic and circular derivatives. Finally, memorise the logarithmic forms of inverse functions—they often appear in exact value questions.
应用奥斯本法则时要小心符号。练习区分双曲导数和三角导数。最后，熟记反函数的对数形式——它们经常出现在求精确值的题目中。
With consistent practice, hyperbolic functions can become one of the most predictable and scoring parts of your IGCSE CCEA Maths paper.
通过持续练习，双曲函数可以成为你 IGCSE CCEA 数学试卷中最有把握拿分的部分之一。
Published by TutorHao | IGCSE CCEA Maths Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图