Hyperbolic Functions AS Mathematics Exam Guide | AS 数学：双曲函数考点精讲

📚 Hyperbolic Functions AS Mathematics Exam Guide | AS 数学：双曲函数考点精讲

Hyperbolic functions are essential building blocks in AS Further Mathematics, linking exponential definitions with identities, calculus, and equations. This revision guide breaks down key definitions, graphs, Osborn’s rule, inverse functions, derivatives, integrals, and common exam traps — all strictly aligned to AS-level expectations. Master these topics to confidently tackle questions on solving hyperbolic equations, proving identities, and integrating hyperbolic expressions.

双曲函数是 AS 进阶数学的重要基础，它将指数定义与恒等式、微积分和方程求解紧密联系起来。本文系统梳理双曲函数的定义、图像、奥斯本法则、反函数、导数和积分，并紧扣 AS 考试要求，帮你从容应对双曲方程求解、恒等式证明以及双曲表达式的积分。

1. Definitions of sinh, cosh and tanh | 双曲正弦、余弦和正切的定义

Hyperbolic sine and cosine are defined via the exponential function. For any real number x,

双曲正弦和双曲余弦通过指数函数定义。对任意实数 x，

sinh x = (eˣ − e⁻ˣ) / 2

cosh x = (eˣ + e⁻ˣ) / 2

From these, the hyperbolic tangent follows naturally as the ratio of sinh to cosh:

由此，双曲正切自然地定义为 sinh 与 cosh 的比值：

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

Notice the similarity to trigonometric functions, but without the imaginary unit. Also, cosh x is always ≥ 1, and cosh 0 = 1, sinh 0 = 0, tanh 0 = 0. The domain of all three is all real numbers, while the range of sinh is ℝ, cosh is [1, ∞), and tanh is (−1, 1).

注意它们与三角函数的相似性，但没有虚数单位。此外，cosh x 始终 ≥ 1，且 cosh 0 = 1，sinh 0 = 0，tanh 0 = 0。三者的定义域均为全体实数，而值域分别为：sinh 是 ℝ，cosh 是 [1, ∞)，tanh 是 (−1, 1)。

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

Graphical interpretation is crucial for visualising domains, ranges and symmetries. Below is a summary of the key features:

图像理解对于把握定义域、值域和对称性至关重要。以下是关键特征总结：

Function	Domain	Range	Symmetry	Asymptotes
sinh x	ℝ	ℝ	Odd	None
cosh x	ℝ	[1, ∞)	Even	None
tanh x	ℝ	(−1, 1)	Odd	y = −1, y = 1

sinh x passes through the origin with increasing gradient. cosh x is a catenary-shaped curve with minimum at (0,1). tanh x approaches horizontal asymptotes at y = ±1 as x → ±∞. Their shapes help when solving inequalities or locating intersections.

sinh x 过原点，且梯度递增。cosh x 呈悬链线形状，最小值在 (0,1)。tanh x 当 x → ±∞ 时趋近于水平渐近线 y = ±1。这些形状有助于求解不等式或确定交点。

3. Osborn’s Rule and Key Hyperbolic Identities | 奥斯本法则与核心双曲恒等式

Osborn’s rule states that any trigonometric identity involving a product of two sines can be converted to a hyperbolic identity by replacing sin with i sinh, cos with cosh, and then removing any factor of i² = −1. Practically, this means:

奥斯本法则指出：任何包含两个正弦乘积的三角恒等式，可通过将 sin 换成 i sinh、cos 换成 cosh，然后消除 i² = −1 的因子，转化为双曲恒等式。实际操作中：

cosh² x − sinh² x = 1

This comes from cos² θ + sin² θ = 1, but note the sign change. Other vital identities include:

这源自 cos² θ + sin² θ = 1，但注意符号变化。其他重要恒等式有：

sinh 2x = 2 sinh x cosh x — analogous to sin 2θ.
cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x — note all plus signs.
tanh² x + sech² x = 1 (where sech x = 1 / cosh x).

等价中文：

sinh 2x = 2 sinh x cosh x
cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x，注意均为加号。
tanh² x + sech² x = 1（其中 sech x = 1 / cosh x）。

Memorise these directly or derive them quickly from exponential definitions. The sign differences from trigonometric identities are a common source of mistakes.

可直接记忆或从指数定义快速推导。与三角恒等式的符号差异是常见错误来源。

4. Solving Equations Involving Hyperbolic Functions | 涉及双曲函数的方程求解

To solve an equation like 5 sinh x + 3 cosh x = 4, express sinh and cosh in terms of exponentials:

对于如 5 sinh x + 3 cosh x = 4 的方程，将 sinh 和 cosh 用指数表示：

5(eˣ − e⁻ˣ)/2 + 3(eˣ + e⁻ˣ)/2 = 4

Multiply by 2: 5(eˣ − e⁻ˣ) + 3(eˣ + e⁻ˣ) = 8, which simplifies to 8eˣ − 2e⁻ˣ = 8. Multiply through by eˣ to get a quadratic: 8e²ˣ − 8eˣ − 2 = 0. Solve for eˣ, then take the natural logarithm to find x. Always check for extraneous solutions.

两边乘 2：5(eˣ − e⁻ˣ) + 3(eˣ + e⁻ˣ) = 8，化简得 8eˣ − 2e⁻ˣ = 8。再同乘 eˣ 得到二次方程 8e²ˣ − 8eˣ − 2 = 0。解出 eˣ 后取自然对数即得 x。务必检验是否有增根。

Another common type is sinh x = a constant, for instance sinh x = 3. Using the exponential form gives eˣ − e⁻ˣ = 6, leading to e²ˣ − 6eˣ − 1 = 0. Then eˣ = 3 + √10 (reject negative exponential). Thus x = ln(3 + √10). This often connects to inverse hyperbolic functions.

另一常见类型为 sinh x = 常数，例如 sinh x = 3。利用指数形式得 eˣ − e⁻ˣ = 6，得 e²ˣ − 6eˣ − 1 = 0，则 eˣ = 3 + √10（舍去负指数），从而 x = ln(3 + √10)。这往往与反双曲函数关联。

5. Inverse Hyperbolic Functions | 反双曲函数

The inverse hyperbolic functions are denoted arsinh, arcosh and artanh, or equivalently sinh⁻¹, cosh⁻¹ and tanh⁻¹. They are defined as follows (with domain restrictions for arcosh and artanh):

反双曲函数记为 arsinh、arcosh 和 artanh，也可写作 sinh⁻¹、cosh⁻¹ 和 tanh⁻¹。它们的定义如下（反双曲余弦和反正切有定义域限制）：

sinh⁻¹ x has domain ℝ and range ℝ.
cosh⁻¹ x has domain [1, ∞) and range [0, ∞).
tanh⁻¹ x has domain (−1, 1) and range ℝ.

中文：sinh⁻¹ x 定义域 ℝ，值域 ℝ；cosh⁻¹ x 定义域 [1, ∞)，值域 [0, ∞)；tanh⁻¹ x 定义域 (−1, 1)，值域 ℝ。

In exam problems, you might need to simplify expressions like cosh(sinh⁻¹ x). Use identities: let y = sinh⁻¹ x, so sinh y = x, then cosh y = √(1 + sinh² y) = √(1 + x²). Always ensure the positive root for cosh⁻¹ range.

考试中常需化简如 cosh(sinh⁻¹ x) 的表达式。令 y = sinh⁻¹ x，则 sinh y = x，于是 cosh y = √(1 + sinh² y) = √(1 + x²)。根据值域始终取正平方根。

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

A crucial skill is expressing inverse hyperbolic functions in terms of natural logarithms. These derivations start from y = sinh⁻¹ x ⇒ x = (eʸ − e⁻ʸ)/2, leading to a quadratic in eʸ.

将反双曲函数用自然对数表示是一项关键技能。推导从 y = sinh⁻¹ x ⇒ x = (eʸ − e⁻ʸ)/2 开始，得到关于 eʸ 的二次方程。

sinh⁻¹ x = ln(x + √(x² + 1))

cosh⁻¹ x = ln(x + √(x² − 1)) (x ≥ 1)

tanh⁻¹ x = ½ ln((1 + x)/(1 − x)) (|x| < 1)

These logarithmic forms are essential for differentiation from first principles and for evaluating exact values. For example, sinh⁻¹ 1 = ln(1 + √2).

这些对数形式对利用定义求导以及计算精确值至关重要。例如，sinh⁻¹ 1 = ln(1 + √2)。

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

Differentiation of hyperbolic functions is straightforward and cyclic, similar to trigonometric derivatives but with sign differences.

双曲函数的微分简单且有周期性，类似于三角函数的导数，但符号有所不同。

d/dx (sinh x) = cosh x

d/dx (cosh x) = sinh x

d/dx (tanh x) = sech² x = 1 − tanh² x

Also useful: d/dx (sech x) = −sech x tanh x, d/dx (cosech x) = −cosech x coth x, d/dx (coth x) = −cosech² x. The chain rule often applies, e.g., d/dx (sinh 3x) = 3 cosh 3x.

此外，有用的还有：d/dx (sech x) = −sech x tanh x，d/dx (cosech x) = −cosech x coth x，d/dx (coth x) = −cosech² x。链式法则常被用到，如 d/dx (sinh 3x) = 3 cosh 3x。

These can be proved using exponential definitions or by the quotient rule starting from sinh/cosh. Practice differentiation alongside implicit and parametric equations involving hyperbolic functions.

这些可用指数定义证明，或从 sinh/cosh 出发用商法则推导。练习时需结合隐函数和参数方程中涉及双曲函数的微分。

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

Differentiating inverse hyperbolic functions yields algebraic results. The standard derivatives are:

对反双曲函数求导将得到代数形式的结果。标准导数如下：

d/dx (sinh⁻¹ x) = 1 / √(x² + 1)

d/dx (cosh⁻¹ x) = 1 / √(x² − 1) (x > 1)

d/dx (tanh⁻¹ x) = 1 / (1 − x²) (|x| < 1)

These can be derived by implicit differentiation (let y = sinh⁻¹ x ⇒ sinh y = x ⇒ cosh y dy/dx = 1) and using cosh y = √(1 + x²). They are frequently tested with integration later.

这些可通过隐函数微分推导（令 y = sinh⁻¹ x ⇒ sinh y = x ⇒ cosh y dy/dx = 1），并利用 cosh y = √(1 + x²)。它们经常在后续的积分问题中出现。

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

Integration reverses differentiation. The basic integrals are:

积分是微分的逆运算。基本积分公式有：

∫ sinh x dx = cosh x + C

∫ cosh x dx = sinh x + C

∫ tanh x dx = ln(cosh x) + C

For definite integrals, you will often combine these with exponential substitutions or use identities. For instance, ∫ sinh² x dx can be tackled using cosh 2x = 1 + 2 sinh² x, giving sinh² x = (cosh 2x − 1)/2.

对于定积分，常需结合指数代换或使用恒等式。例如，∫ sinh² x dx 可利用 cosh 2x = 1 + 2 sinh² x，得到 sinh² x = (cosh 2x − 1)/2 进行积分。

Also useful: ∫ sech² x dx = tanh x + C. When the integrand is a product, consider integration by parts or a substitution u = eˣ when exponentials simplify the expression.

同样有用的还有 ∫ sech² x dx = tanh x + C。当被积函数为乘积时，可考虑分部积分，或当指数可简化表达式时代换 u = eˣ。

10. Integrals Leading to Inverse Hyperbolic Functions | 导出反双曲函数的积分

Some standard integrals yield inverse hyperbolic functions. Recognising these forms saves time in exams.

某些标准积分结果会得到反双曲函数。识别这些形式可以节省考试时间。

∫ 1 / √(x² + a²) dx = sinh⁻¹ (x / a) + C (a > 0)

∫ 1 / √(x² − a²) dx = cosh⁻¹ (x / a) + C (x > a > 0)

∫ 1 / (a² − x²) dx = (1/a) tanh⁻¹ (x / a) + C (|x| < a)

These are closely related to the logarithmic forms. For example, ∫ dx / √(x² + 4) = sinh⁻¹ (x/2) + C. Make sure you choose the correct inverse function based on the integrand’s structure and domain.

它们与对数形式密切相关。例如，∫ dx / √(x² + 4) = sinh⁻¹ (x/2) + C。务必根据被积函数的结构和定义域选择正确的反函数。

11. Exam Tips for Hyperbolic Functions | 双曲函数考试技巧

Check signs carefully: the identity cosh² x − sinh² x = 1 has a minus sign, unlike the trigonometric version. Many students lose marks by writing cosh² x + sinh² x = 1. — 仔细检查符号：恒等式 cosh² x − sinh² x = 1 有减号，与三角版本不同。很多学生误写成 cosh² x + sinh² x = 1 而丢分。
Handle exponentials systematically: when solving equations, always multiply by eˣ to form a quadratic in eˣ. Reject any negative values for eˣ. — 系统处理指数：解方程时，始终同乘 eˣ 构造关于 eˣ 的二次方程。舍去 eˣ 的任何负值。
Know your ranges: cosh x ≥ 1 informs why arcosh x domain is [1, ∞). tanh x ∈ (−1,1) aids in finding limits. — 熟悉值域：cosh x ≥ 1 决定了 arcosh x 的定义域为 [1, ∞)。tanh x ∈ (−1,1) 有助于求极限。
Derivatives of inverse functions: memorise the standard forms, but be able to derive them quickly using implicit differentiation. — 反函数的导数：记住标准形式，但也要能快速用隐函数微分推导。
Integration by recognition: practice spotting integrals like ∫ 3/√(9x²+16) dx that lead to sinh⁻¹. — 识别积分类型：练习识别形如 ∫ 3/√(9x²+16) dx 的积分以得到 sinh⁻¹。

Always show your substitution or rearrangement steps explicitly — method marks are generous in AS hyperbolic function questions.

务必清晰展示代换或整理步骤——AS 双曲函数题目中步骤分相当可观。

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Hyperbolic Functions AS Mathematics Exam Guide | AS 数学：双曲函数 考点精讲