Hyperbolic Functions | 双曲函数考点精讲

📚 Hyperbolic Functions | 双曲函数考点精讲

Hyperbolic functions are a family of exponential functions that share many properties with trigonometric functions, yet they are defined using the exponential function eˣ rather than the unit circle. In A-Level WJEC Mathematics, you are expected to understand their definitions, sketch their graphs, derive and apply identities, differentiate and integrate them, and handle their inverses. This article covers every essential topic in fine detail, pairing each explanation in English and Chinese to support bilingual learners.

双曲函数是一组与三角函数有许多相似性质的指数型函数，不过它们是基于指数函数 eˣ 而非单位圆来定义的。在 A-Level WJEC 数学中，你需要掌握双曲函数的定义、绘制图像、推导并应用恒等式、求导与积分，以及处理反双曲函数。本文细致讲解每一个核心考点，每一段英文与中文配对呈现，帮助双语学习者准确理解。

1. Definitions of Hyperbolic Functions | 双曲函数的定义

The two fundamental hyperbolic functions, sinh x and cosh x, are defined in terms of the exponential function. Hyperbolic sine is given by sinh x = (eˣ − e⁻ˣ)/2, and hyperbolic cosine by cosh x = (eˣ + e⁻ˣ)/2. From these, tanh x is defined as sinh x / cosh x = (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ). Their domain is all real numbers, and they are useful for describing catenaries, rapid exponential growth, and solutions to certain differential equations.

双曲函数最基本的两个函数 sinh x 和 cosh x 都通过指数函数定义。双曲正弦为 sinh x = (eˣ − e⁻ˣ)/2，双曲余弦为 cosh x = (eˣ + e⁻ˣ)/2。由此，tanh x 定义为 sinh x / cosh x = (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ)。它们的定义域均为全体实数，常用于描述悬链线、快速指数增长以及某些微分方程的解。

sinh x = (eˣ − e⁻ˣ)/2
cosh x = (eˣ + e⁻ˣ)/2
tanh x = sinh x / cosh x

Corresponding reciprocal functions are also defined: cosech x = 1/sinh x, sech x = 1/cosh x, coth x = 1/tanh x = cosh x / sinh x. These appear less frequently but are part of the specification.

对应的倒数函数也有定义：cosech x = 1/sinh x，sech x = 1/cosh x，coth x = 1/tanh x = cosh x / sinh x。它们出现频率较低，但仍在考纲范围内。

2. Graphs of Hyperbolic Functions | 双曲函数的图像

The graph of y = sinh x is an odd function that passes through the origin with gradient 1. It grows like ½eˣ as x → ∞ and like −½e⁻ˣ as x → −∞. The graph of y = cosh x is an even function with a minimum point at (0,1), shaped like a symmetric hanging chain. It grows like ½eˣ for both large positive and negative x. The graph of y = tanh x is an odd function with horizontal asymptotes at y = ±1, passing through the origin with gradient 1.

y = sinh x 的图像是一个奇函数，经过原点且斜率为 1。当 x → ∞ 时，图像形如 ½eˣ；当 x → −∞ 时，形如 −½e⁻ˣ。y = cosh x 是偶函数，最低点为 (0,1)，形状像一条对称的悬挂链条，在 x 正负无穷大时都像 ½eˣ 一样增长。y = tanh x 是奇函数，有水平渐近线 y = ±1，经过原点且斜率为 1。

sinh x: odd, passes through (0,0), no asymptotes.
cosh x: even, minimum at (0,1), looks like a U-shape but flatter near the vertex.
tanh x: odd, asymptotes y = 1 and y = −1, range (−1,1).

These graphs are often assessed in sketching questions, where you may need to label intercepts, asymptotes, and the general shape.

这些图像经常在作图题中考查，你需要标出截距、渐近线以及大致形状。

3. Hyperbolic Identities | 双曲恒等式

Hyperbolic functions satisfy identities that closely mirror trigonometric ones, but with sign changes. The fundamental identity is cosh²x − sinh²x = 1. Compare this with cos²x + sin²x = 1. From this, other identities can be derived, such as 1 − tanh²x = sech²x and coth²x − 1 = cosech²x.

双曲函数满足与三角函数极为相似的恒等式，但符号有所变化。最基本的恒等式是 cosh²x − sinh²x = 1，对比三角函数的 cos²x + sin²x = 1。由此可以推导出其他形式，例如 1 − tanh²x = sech²x 以及 coth²x − 1 = cosech²x。

Double argument formulas also exist: sinh(2x) = 2 sinh x cosh x, and cosh(2x) = cosh²x + sinh²x = 2 cosh²x − 1 = 1 + 2 sinh²x. These are useful when solving equations or integrating.

倍角公式也存在：sinh(2x) = 2 sinh x cosh x，cosh(2x) = cosh²x + sinh²x = 2 cosh²x − 1 = 1 + 2 sinh²x。这些在解方程或积分时很有用。

4. Osborne’s Rule | 奥斯本法则

Osborne’s rule provides a quick way to convert any trigonometric identity into its hyperbolic counterpart: replace each trigonometric function with the corresponding hyperbolic function, and change the sign of any product (or implied product) of two sines. This is because sin²x appears with a positive sign in trig identities, while sinh²x appears with a negative sign in the fundamental hyperbolic identity.

奥斯本法则提供了一种将三角恒等式快速转化为双曲恒等式的方法：将每一个三角函数替换为对应的双曲函数，并将任何两个正弦的乘积（或隐含的乘积）的符号改变。这是因为在三角恒等式中 sin²x 带正号，而在双曲基本恒等式中 sinh²x 带负号。

For example, starting from cos(A−B) = cos A cos B + sin A sin B, applying Osborne’s rule gives cosh(A−B) = cosh A cosh B − sinh A sinh B. Always verify with the original exponential definitions if unsure.

例如，从 cos(A−B) = cos A cos B + sin A sin B 出发，应用奥斯本法则得到 cosh(A−B) = cosh A cosh B − sinh A sinh B。如果不确定，请用原始指数定义进行验证。

5. Derivatives of Hyperbolic Functions | 双曲函数的导数

The derivatives of hyperbolic functions are straightforward and do not involve sign changes like their trigonometric counterparts. The derivative of sinh x is cosh x. The derivative of cosh x is sinh x. The derivative of tanh x is sech²x. These can be derived from the exponential definitions or the quotient rule.

双曲函数的求导非常直接，不像三角函数那样有符号变化。sinh x 的导数是 cosh x，cosh x 的导数是 sinh x，tanh x 的导数是 sech²x。这些结果可以用指数定义或商法则推导得出。

d/dx (sinh x) = cosh x
d/dx (cosh x) = sinh x
d/dx (tanh x) = sech²x

When dealing with composite functions, use the chain rule. For example, d/dx [sinh(ax+b)] = a cosh(ax+b). A common exam question is to differentiate expressions like ln(cosh x), which yields tanh x, because d/dx [cosh x] = sinh x and then 1/cosh x times sinh x gives tanh x.

处理复合函数时运用链式法则。例如 d/dx [sinh(ax+b)] = a cosh(ax+b)。常见的考试题是求 ln(cosh x) 的导数，得到 tanh x，因为 d/dx [cosh x] = sinh x，然后乘以 1/cosh x 得到 tanh x。

6. Integrals of Hyperbolic Functions | 双曲函数的积分

Since differentiation and integration are reverse processes, the integrals follow directly. ∫ sinh x dx = cosh x + C, ∫ cosh x dx = sinh x + C, and ∫ sech²x dx = tanh x + C. Slightly more involved is ∫ tanh x dx = ln(cosh x) + C, which is obtained by writing tanh x as sinh x / cosh x and recognising the derivative of the denominator.

由于积分是求导的逆运算，积分公式可以直接得出。∫ sinh x dx = cosh x + C，∫ cosh x dx = sinh x + C，∫ sech²x dx = tanh x + C。稍复杂的是 ∫ tanh x dx = ln(cosh x) + C，可以将 tanh x 写成 sinh x / cosh x 并识别出分母的导数来得到。

For definite integrals, you may be asked to find areas under hyperbolic curves. For instance, the area under y = sinh x from 0 to a is cosh a − 1. Always remember to add the constant of integration for indefinite integrals, and use exact values where possible.

对于定积分，你可能需要计算双曲曲线下的面积。比如 y = sinh x 在 0 到 a 区间下的面积为 cosh a − 1。不定积分一定要记得加上积分常数，并尽可能使用精确值。

7. Inverse Hyperbolic Functions | 反双曲函数

The inverse hyperbolic functions are denoted as arsinh x, arcosh x, and artanh x. Some textbooks use the notation sinh⁻¹ x, but this is not a reciprocal. Their domains and ranges must be learned: arsinh x has domain ℝ and range ℝ; arcosh x has domain x ≥ 1 and range y ≥ 0; artanh x has domain |x| < 1 and range ℝ.

反双曲函数记作 arsinh x, arcosh x, artanh x。有些教材也写成 sinh⁻¹ x，但这并不是倒数。它们的定义域和值域必须掌握：arsinh x 的定义域是 ℝ，值域也是 ℝ；arcosh x 的定义域是 x ≥ 1，值域是 y ≥ 0；artanh x 的定义域是 |x| < 1，值域是 ℝ。

A common exam task is to sketch these inverses by reflecting the graphs of the original functions in the line y = x. For arcosh x, only the positive branch of cosh x is used to ensure it is one-to-one. The graph of arsinh x passes through the origin and looks similar to a logarithmic curve.

常见的考题是将原函数的图像关于直线 y = x 反射得到反函数的图像。对于 arcosh x，只使用 cosh x 的正分支，以保证一一对应。arsinh x 的图像过原点，形状与对数曲线相似。

8. Logarithmic Form of Inverse Hyperbolic Functions | 反双曲函数的对数形式

Each inverse hyperbolic function can be expressed in logarithmic form, which is particularly useful for solving equations without a calculator. These forms are derived by letting y = arsinh x, writing x = sinh y in exponential form, and solving for y. The results are:

每个反双曲函数都可以用对数形式表示，这在没有计算器时解方程特别有用。这些形式可以通过设 y = arsinh x，将 x = sinh y 写成指数形式并解出 y 而得到。结果为：

arsinh x = ln(x + √(x²+1))
arcosh x = ln(x + √(x²−1)), x ≥ 1
artanh x = ½ ln((1+x)/(1−x)), |x| < 1

You should be able to prove these derivations step by step. For example, to derive arsinh x, set y = arsinh x ⇒ sinh y = x ⇒ (eʸ − e⁻ʸ)/2 = x. Multiply by 2eʸ and rearrange to a quadratic in eʸ: e²ʸ − 2x eʸ − 1 = 0. Solving gives eʸ = x + √(x²+1) (since eʸ > 0). Taking natural logs yields the result.

你应该能够逐步推导这些形式。例如，推导 arsinh x：设 y = arsinh x ⇒ sinh y = x ⇒ (eʸ − e⁻ʸ)/2 = x。两边乘 2eʸ 并整理成关于 eʸ 的二次方程：e²ʸ − 2x eʸ − 1 = 0。解得 eʸ = x + √(x²+1)（因 eʸ > 0）。取自然对数即得结果。

9. Derivatives of Inverse Hyperbolic Functions | 反双曲函数的导数

The derivatives of inverse hyperbolic functions are algebraic, making them important for integration later. They are: d/dx (arsinh x) = 1/√(x²+1), d/dx (arcosh x) = 1/√(x²−1) for x > 1, and d/dx (artanh x) = 1/(1−x²) for |x| < 1.

反双曲函数的导数是代数形式的，这使它们在后续积分中非常重要。它们分别是：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)（|x| < 1）。

These results can be proven using implicit differentiation. For y = arsinh x, we have sinh y = x, so cosh y (dy/dx) = 1. But cosh y = √(1+sinh²y) = √(1+x²), so dy/dx = 1/√(1+x²). Similar techniques work for the others. Integrating these standard forms in reverse is a key skill, linking directly to the next topic.

这些结果可以用隐函数求导证明。对于 y = arsinh x，有 sinh y = x，于是 cosh y (dy/dx) = 1。而 cosh y = √(1+sinh²y) = √(1+x²)，故 dy/dx = 1/√(1+x²)。其他情况类似。反过来对这些标准形式进行积分是一项关键技能，直接联系到下一个主题。

10. Solving Equations with Hyperbolic Functions | 解含双曲函数的方程

Equations involving hyperbolic functions usually reduce to quadratics in eˣ or sinh x/cosh x. For example, to solve 3 cosh x − 2 sinh x = 5, express cosh x and sinh x in terms of eˣ and e⁻ˣ, multiply by 2eˣ, and solve the resulting quadratic in eˣ. After finding eˣ, take natural logs to obtain x. Always check that solutions are valid in the original equation.

涉及双曲函数的方程通常可化为关于 eˣ 或 sinh x/cosh x 的二次方程。例如，要解 3 cosh x − 2 sinh x = 5，将 cosh x 和 sinh x 用 eˣ、e⁻ˣ 表示，两边乘以 2eˣ，然后解关于 eˣ 的二次方程。求得 eˣ 后，取自然对数得到 x。务必验证解在原始方程中有效。

Another common type is to use identities. For instance, to solve cosh²x − sinh x = 1, replace cosh²x with 1 + sinh²x, yielding a quadratic in sinh x. After solving for sinh x, use arsinh or logarithmic forms to find x.

另一种常见类型是使用恒等式。例如，解 cosh²x − sinh x = 1 时，用 1 + sinh²x 替换 cosh²x，得到关于 sinh x 的二次方程。解出 sinh x 后，用 arsinh 或对数形式求出 x。

Sometimes you need to solve equations like tanh x = 0.5. Here, rewriting tanh x as (e²ˣ − 1)/(e²ˣ + 1) and cross-multiplying leads to e²ˣ = 3, so x = ½ ln 3. Alternatively, use artanh directly, but many marks are awarded for algebraic manipulation from the exponential definitions.

有时需解如 tanh x = 0.5 的方程。将 tanh x 写成 (e²ˣ − 1)/(e²ˣ + 1)，交叉相乘得 e²ˣ = 3，因此 x = ½ ln 3。另一种方式是直接用 artanh，但通常用指数定义的代数推导能获得更多步骤分。

11. Applications: The Catenary | 应用：悬链线

A classic application of hyperbolic functions is the catenary, the shape of a hanging flexible chain or cable under its own weight. Its equation is y = a cosh(x/a), where a is a constant. The curve is symmetric, with its lowest point at x = 0. The slope of the catenary at any point is given by dy/dx = sinh(x/a), and the arc length can be expressed elegantly using hyperbolic functions.

双曲函数的一个经典应用是悬链线，即柔软链条或缆绳在自身重量下悬挂所形成的形状。其方程为 y = a cosh(x/a)，其中 a 为常数。曲线对称，最低点在 x = 0 处。悬链线上任一点的斜率为 dy/dx = sinh(x/a)，弧长也可以用双曲函数优雅地表达。

In WJEC exams, questions may provide a context involving a hanging cable and ask you to find the height at a specific point, the tension, or the length. You will need to substitute values into the cosh function and interpret the result. Though not overly frequent, such application questions differentiate higher-grade candidates.

在 WJEC 考试中，可能会出现悬挂缆绳的情景题，要求你计算某点的高度、张力或长度。你需要将数值代入 cosh 函数并解释结果。虽然这类应用题不算频繁，但能区分高分段考生。

12. Exam Tips and Common Pitfalls | 考试技巧与常见误区

Always remember that cosh x ≥ 1 for all real x – an easy mark but often overlooked. When differentiating, do not be tempted to add a minus sign; cosh x differentiates to sinh x without any sign change. When integrating, treat hyperbolic functions like exponential functions, and always check if the result differentiates back correctly.

始终牢记 cosh x ≥ 1 对所有实数 x 成立——这是一个容易拿分却常被忽略的点。在求导时，不要习惯性地加负号；cosh x 的导数是 sinh x，没有符号变化。积分时，将双曲函数视为指数函数处理，并始终检查求导后能否回到原函数。

Be meticulous with the domain of arcosh and artanh. Many students mistakenly apply arcosh to values less than 1. When solving quadratics in eˣ, discard any negative roots since eˣ > 0. In identity proofs, starting from the more complex side and using exponential definitions is usually safest.

小心 arcosh 和 artanh 的定义域。许多学生错误地将 arcosh 应用于小于 1 的值。在解关于 eˣ 的二次方程时，舍去任何负根，因为 eˣ > 0。在证明恒等式时，从较复杂的一侧开始并使用指数定义通常是最稳妥的方法。

Finally, practise sketching graphs with labelled axes, asymptotes, and intercepts. Marks are often allocated for these details. Linking hyperbolic and inverse hyperbolic graphs through reflection can also help solidify understanding.

最后，多加练习绘制带有标注坐标轴、渐近线和截距的图像。这些细节往往是采分点。通过反射关系联系双曲函数与反双曲函数的图像，也有助于巩固理解。

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Hyperbolic Functions | 双曲函数考点精讲