Hyperbolic Functions Exam Focus | A-Level 数学：双曲函数考点精讲

📚 Hyperbolic Functions Exam Focus | A-Level 数学：双曲函数考点精讲

Hyperbolic functions – sinh, cosh, tanh and their inverses – often appear in A-Level Mathematics as an extension of calculus and algebraic manipulation. Understanding their definitions, identities, derivatives and integrals is essential for mastering the syllabus and tackling exam questions with confidence. This article provides a structured revision guide covering every key topic.

双曲函数——sinh、cosh、tanh 及其反函数——在 A-Level 数学中经常作为微积分和代数操作的延伸出现。掌握它们的定义、恒等式、导数和积分，对于把握考纲并自信地解答考题至关重要。本文提供一份结构化的复习指南，涵盖每一个关键主题。

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

The hyperbolic sine and cosine are defined in terms of exponential functions:

双曲正弦与双曲余弦用指数函数定义：

sinh x = (eˣ − e⁻ˣ) / 2

cosh x = (eˣ + e⁻ˣ) / 2

From these, we define tanh x, sech x, coth x and cosech x in an analogous way to trigonometric ratios.

由此可以类似三角函数定义 tanh x、sech x、coth x 和 cosech x。

tanh x = sinh x / cosh x

Note that cosh x is always ≥ 1, and sinh x has the same sign as x and passes through the origin.

注意 cosh x 始终 ≥ 1，而 sinh x 与 x 同号且过原点。

2. Graphs and Basic Properties | 图像与基本性质

The graph of y = sinh x is an odd function with a single increasing curve passing through (0,0). It closely resembles y = ½eˣ for large positive x and y = −½e⁻ˣ for large negative x.

y = sinh x 的图像是奇函数，单增曲线过 (0,0)。当 x 很大时它近似 y = ½eˣ，而 x 很小时近似 y = −½e⁻ˣ。

The graph of y = cosh x is an even function with its minimum at (0,1) and rises symmetrically on both sides. It is sometimes called a catenary.

y = cosh x 的图像是偶函数，最低点为 (0,1)，两侧对称上升。它有时被称为悬链线。

y = tanh x is an odd function bounded between horizontal asymptotes y = 1 and y = −1. Its graph is S-shaped, passing through the origin.

y = tanh x 是奇函数，有两条水平渐近线 y = 1 和 y = −1。图像呈 S 形，经过原点。

3. Osborne’s Rule and Identities | 奥斯本规则与恒等式

Most trigonometric identities can be converted into hyperbolic identities by replacing sin with i sinh and cos with cosh, then simplifying. Osborne’s Rule: change the sign of a term whenever a product (or implied product) of two sines occurs.

大多数三角恒等式都可以转化为双曲恒等式，只需将 sin 替换为 i sinh、cos 替换为 cosh 然后化简。奥斯本规则：每当出现两个正弦的乘积（或隐含乘积）项时，改变该项的符号。

The fundamental identity is:

基本恒等式为：

cosh² x − sinh² x = 1

Dividing by cosh² x gives: 1 − tanh² x = sech² x. Similarly, dividing by sinh² x gives: coth² x − 1 = cosech² x.

除以 cosh² x 得到 1 − tanh² x = sech² x。除以 sinh² x 得到 coth² x − 1 = cosech² x。

Compare with the trigonometric versions: cos² x + sin² x = 1, 1 + tan² x = sec² x, cot² x + 1 = cosec² x. Notice the sign differences.

与三角恒等式比较：cos² x + sin² x = 1，1 + tan² x = sec² x，cot² x + 1 = cosec² x。注意符号差异。

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

The derivatives of the basic hyperbolic functions are straightforward and very similar to their trigonometric counterparts, but without the sign changes.

基本双曲函数的导数很直接，非常类似于它们的三角对应，但没有符号变化。

d/dx (sinh x) = cosh x

d/dx (cosh x) = sinh x

d/dx (tanh x) = sech² x

For composite functions, use the chain rule. Example: d/dx (sinh(2x)) = 2 cosh(2x).

对于复合函数，使用链式法则。例如：d/dx (sinh(2x)) = 2 cosh(2x)。

Derivatives of sech x, coth x and cosech x follow from these definitions and quotient/product rules.

sech x、coth x 和 cosech x 的导数可由商法则或积法则推出。

5. Integrals Involving Hyperbolic Functions | 涉及双曲函数的积分

Since differentiation and integration are inverse processes, the standard integrals are immediate:

由于微分和积分互为逆运算，标准积分直接可得：

∫ sinh x dx = cosh x + C

∫ cosh x dx = sinh x + C

∫ tanh x dx = ln(cosh x) + C

For integrals like ∫ sinh(ax+b) dx, use the reverse chain rule: the result is (1/a) cosh(ax+b) + C.

对于诸如 ∫ sinh(ax+b) dx 的积分，使用逆链式法则：结果为 (1/a) cosh(ax+b) + C。

Integrals requiring completion of the square or substitution often lead to inverse hyperbolic forms, which are covered later.

需要配平方或替换的积分常得到反双曲函数形式，稍后介绍。

6. Inverse Hyperbolic Functions | 反双曲函数

The inverse functions are written as arsinh x, arcosh x, artanh x (or sometimes sinh⁻¹, cosh⁻¹, tanh⁻¹).

反函数写作 arsinh x、arcosh x、artanh x（有时也写作 sinh⁻¹、cosh⁻¹、tanh⁻¹）。

Domain considerations: arsinh x is defined for all real x; arcosh x is defined for x ≥ 1; artanh x is defined for |x| < 1.

定义域考虑：arsinh x 对所有实数 x 有定义；arcosh x 定义域为 x ≥ 1；artanh x 定义域为 |x| < 1。

Their graphs are reflections of the original hyperbolic graphs in the line y = x, suitably restricted for arcosh.

它们的图像是原双曲函数图像关于直线 y = x 的反射，对于 arcosh 要适当限制。

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

Each inverse hyperbolic function can be expressed in terms of natural logarithms. These forms are extremely useful for differentiation and integration.

每个反双曲函数都可以用自然对数表示。这些形式对于微积分极其有用。

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 are derived by writing y = sinh⁻¹ x ⇒ x = sinh y, expressing in exponentials and solving the resulting quadratic in eʸ.

这些形式通过令 y = sinh⁻¹ x ⇒ x = sinh y，用指数表示并求解关于 eʸ 的二次方程得到。

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

The derivatives are found either by differentiating the logarithmic forms directly or by implicit differentiation. The results are rational or algebraic functions:

导数可通过直接微分对数形式或隐微分求得。结果是代数函数或有理函数：

d/dx (arsinh x) = 1 / √(x² + 1)

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

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

These are extremely valuable when integrating functions of the form 1/√(x²±a²) and 1/(a²−x²).

这在积分形如 1/√(x²±a²) 和 1/(a²−x²) 的函数时极为珍贵。

9. Solving Equations with Hyperbolic Functions | 用双曲函数解方程

Equations involving hyperbolic functions are often solved by converting to exponential form or by using identities.

涉及双曲函数的方程常通过转化为指数形式或使用恒等式来求解。

Example: Solve 3 cosh x − 2 sinh x = 4. Substitute the definitions and multiply through by 2 to get 3(eˣ+e⁻ˣ) − 2(eˣ−e⁻ˣ) = 8, which simplifies to eˣ + 5e⁻ˣ = 8. Multiply by eˣ: e²ˣ − 8eˣ + 5 = 0. Solve the quadratic in eˣ; finally take ln to find x.

例子：解方程 3 cosh x − 2 sinh x = 4。代入定义并乘以 2 得到 3(eˣ+e⁻ˣ) − 2(eˣ−e⁻ˣ) = 8，化简得 eˣ + 5e⁻ˣ = 8。乘以 eˣ：e²ˣ − 8eˣ + 5 = 0。解关于 eˣ 的二次方程，最后取 ln 得到 x。

Equations can also be solved using Osborne’s rule to convert a hyperbolic equation into a trigonometric one, solve, then convert back, though this is less common.

也可用奥斯本规则将双曲方程转化为三角方程，求解后再转回，但这种方法不常见。

10. Key Exam Tips | 关键解题技巧

Memorise the fundamental identity and derivatives accurately. Confusing the sign differences with trigonometric derivatives is a common mistake.

准确记忆基本恒等式和导数。与三角导数的符号混淆是常见错误。

When integrating, always check if a fraction matches the standard forms for inverse hyperbolic functions. Recognising patterns like ∫ dx/√(x²+4) saves time.

积分时，务必检查分式是否匹配反双曲函数的标准形式。识别出像 ∫ dx/√(x²+4) 这样的模式可以节省时间。

For proofs, start from the exponential definitions or apply identities logically. Keep a table of logarithmic forms handy during revision.

对于证明题，从指数定义开始，或合乎逻辑地应用恒等式。复习时准备一张对数形式表格会很有帮助。

Mastery of hyperbolic functions also unlocks easier solutions to certain differential equations and integration problems later in the A-Level course.

掌握双曲函数也能在 A-Level 课程的后续部分，更轻松地解决某些微分方程和积分问题。

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Hyperbolic Functions Exam Focus | A-Level 数学：双曲函数 考点精讲