Hyperbolic Functions: AQA Exam Essentials | 双曲函数考点精讲

📚 Hyperbolic Functions: AQA Exam Essentials | 双曲函数考点精讲

Hyperbolic functions appear throughout AQA A Level Further Mathematics, combining exponentials, calculus, and identities in a distinctive way. This guide covers the essential definitions, graphs, differentiation, integration, identities, and exam strategies you need to tackle hyperbolic function questions with confidence.

双曲函数贯穿 AQA A Level 进阶数学的多个模块，以独特方式融合了指数函数、微积分与恒等式。本指南涵盖必考的定义、图像、微分、积分、恒等式及应试策略，助你从容应对双曲函数考题。

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

Hyperbolic sine and cosine are defined using exponentials: sinh x = (eˣ – e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2. The hyperbolic tangent is tanh x = sinh x / cosh x. Their reciprocal functions are cosech x = 1/sinh x, sech x = 1/cosh x, coth x = 1/tanh x. You should know these off by heart, as they underpin every manipulation with hyperbolic functions.

双曲正弦和余弦由指数函数定义：sinh x = (eˣ – e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2。双曲正切为 tanh x = sinh x / cosh x。其倒数为 cosech x = 1/sinh x，sech x = 1/cosh x，coth x = 1/tanh x。这些定义是处理双曲函数的基础，必须牢记。

The exponential definitions immediately reveal the odd/even nature: sinh(-x) = -sinh x, cosh(-x) = cosh x. For any real x, cosh x ≥ 1, while sinh x can take any real value.

由指数定义立即可得奇偶性：sinh(-x) = -sinh x，cosh(-x) = cosh x。对于任意实数 x，cosh x ≥ 1，而 sinh x 可取任意实数值。

2. Graphs and Key Features | 图像与关键特征

The graph of y = sinh x is odd and passes through the origin, growing like eˣ/2 for large positive x and like –e⁻ˣ/2 for large negative x. The graph of y = cosh x is even, symmetric about the y-axis, with a minimum point at (0,1) and grows as eˣ/2 in both directions. The graph of y = tanh x has horizontal asymptotes y = 1 and y = –1, passes through the origin, and its derivative at 0 is 1.

y = sinh x 的图像是奇函数，过原点，x 正向很大时如 eˣ/2 增长，负向很大时如 –e⁻ˣ/2 变化。y = cosh x 的图像是偶函数，关于 y 轴对称，最低点为 (0,1)，两端均按 eˣ/2 增长。y = tanh x 有水平渐近线 y = 1 与 y = –1，过原点，且在 x=0 处导数为 1。

These graphs help visualise domain, range, and the behaviour of combinations such as the catenary y = a cosh(x/a), which describes a hanging chain.

这些图像有助于直观理解定义域、值域，以及如悬链线 y = a cosh(x/a) 等组合函数的行为，后者描述悬挂的链条形状。

3. Inverse Hyperbolic Functions | 反双曲函数

Inverse hyperbolic functions can be expressed in logarithmic form, making them extremely useful for integration and solving equations:

反双曲函数可以写成对数形式，这使其在积分和解方程中极为有用：

arsinh x = ln(x + √(x²+1)), x ∈ ℝ

arcosh x = ln(x + √(x²–1)), x ≥ 1

artanh x = ½ ln((1+x)/(1–x)), |x| < 1

Note the domain restrictions: arcosh x is defined only for x ≥ 1 (two branches, but principal value uses x ≥ 1); artanh x is defined for |x| < 1. For arsinh x there is no restriction — it covers all real x.

注意定义域限制：arcosh x 仅当 x ≥ 1 时有定义（主值支取 x ≥ 1）；artanh x 要求 |x| < 1；arsinh x 无限制，覆盖全体实数。

4. Hyperbolic Identities | 双曲恒等式

The fundamental identity is cosh²x – sinh²x = 1. From this, dividing by cosh²x gives 1 – tanh²x = sech²x, and dividing by sinh²x gives coth²x – 1 = cosech²x.

最基本的恒等式是 cosh²x – sinh²x = 1。将其除以 cosh²x 得到 1 – tanh²x = sech²x，除以 sinh²x 得到 coth²x – 1 = cosech²x。

Compound-angle and double-angle formulas closely mimic trigonometry, but watch the signs:

和角与倍角公式与三角学极为相似，但需留意符号：

sinh(A ± B) = sinhA coshB ± coshA sinhB

cosh(A ± B) = coshA coshB ± sinhA sinhB

sinh 2x = 2 sinh x cosh x

cosh 2x = cosh²x + sinh²x = 2 cosh²x – 1 = 1 + 2 sinh²x

These identities are essential for simplifying expressions and integrating products of hyperbolic functions.

这些恒等式在化简表达式和积分双曲函数乘积时不可或缺。

5. Osborn’s Rule | 奥斯本定则

Osborn’s Rule gives a quick way to convert any trigonometric identity into the corresponding hyperbolic identity: in the trig identity, replace every product (or implied product) of two sines by the negative of the product of two hyperbolic sines, and replace every other trigonometric function directly by its hyperbolic counterpart. No sign changes are needed for cos, tan, sec, csc, cot besides those produced by sines.

奥斯本定则提供了一种将三角恒等式快速转换为双曲恒等式的方法：在三角恒等式中，将每处两个正弦的乘积（包括隐含乘积）替换为相应双曲正弦乘积的相反数，其他三角函数直接对应为双曲函数。除正弦引起的符号变化外，余弦、正切等均无需改变符号。

For example, starting from cos²θ + sin²θ = 1, there is one product of two sines: sin²θ. Replace it with –sinh²x and cos²θ with cosh²x, yielding cosh²x – sinh²x = 1. Similarly, sin 2θ = 2 sin θ cos θ becomes sinh 2x = 2 sinh x cosh x with no sign change because there is only one product of a sine and a cosine — not a product of two sines.

例如，从 cos²θ + sin²θ = 1 出发，有一个正弦乘积 sin²θ，将其替换为 –sinh²x，cos²θ 替换为 cosh²x，即得 cosh²x – sinh²x = 1。类似地，sin 2θ = 2 sin θ cos θ 变成 sinh 2x = 2 sinh x cosh x，无需变号，因为此处只有一个正弦与一个余弦的乘积，而非两个正弦的乘积。

6. Differentiation of Hyperbolic Functions | 双曲函数的微分

The derivatives of the basic hyperbolic functions pair up neatly, but the reciprocal functions carry sign differences compared with their trig cousins:

基本双曲函数的导数成对出现，但倒数双曲函数与对应三角函数的导数在符号上有所不同：

d/dx sinh x = cosh x

d/dx cosh x = sinh x

d/dx tanh x = sech² x

d/dx sech x = –sech x tanh x

d/dx cosech x = –cosech x coth x

d/dx coth x = –cosech² x

Notice how cosh differentiates to sinh without a sign change, unlike cos → –sin. The chain rule extends these straightforwardly, e.g. d/dx sinh(ax+b) = a cosh(ax+b).

注意 cosh 求导得到 sinh，符号不变，与 cos → –sin 不同。结合链式法则可轻松扩展，如 d/dx sinh(ax+b) = a cosh(ax+b)。

7. Differentiation of Inverse Hyperbolic Functions | 反双曲函数的微分

The derivatives of inverse hyperbolic functions are algebraic and appear frequently in integration:

反双曲函数的导数为代数形式，在积分中频繁出现：

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 can be proved by differentiating the logarithmic forms or by using y = arsinh x ⇒ sinh y = x and implicit differentiation with cosh²y – sinh²y = 1.

这些公式可通过对对数形式求导，或设 y = arsinh x ⇒ sinh y = x 并利用 cosh²y – sinh²y = 1 隐函数求导加以证明。

When the argument is a function of x, use the chain rule: d/dx arsinh(2x) = 2/√(4x²+1).

当自变量为 x 的函数时，使用链式法则：d/dx arsinh(2x) = 2/√(4x²+1)。

8. Integration of Hyperbolic Functions | 双曲函数的积分

Basic hyperbolic integrals mirror differentiation results:

基本双曲积分与微分结果对应：

∫ sinh x dx = cosh x + C

∫ cosh x dx = sinh x + C

∫ tanh x dx = ln |cosh x| + C

∫ sech² x dx = tanh x + C

Standard integrals that give inverse hyperbolic functions are particularly useful:

得到反双曲函数的标准积分尤其有用：

∫ 1/√(x²+a²) dx = arsinh(x/a) + C

∫ 1/√(x²–a²) dx = arcosh(x/a) + C, x > a

∫ 1/(a²–x²) dx = (1/a) artanh(x/a) + C, |x| < a

The AQA formula booklet includes these, but you must be able to recognise which form to use and adjust constants accordingly.

AQA 公式手册中包含这些公式，但你必须能够识别使用哪种形式，并据此调整常数。

9. Solving Hyperbolic Equations | 解双曲方程

Many hyperbolic equations can be solved by converting everything to exponentials. For example, 2 sinh x + 3 cosh x = 5 becomes (eˣ – e⁻ˣ) + (3/2)(eˣ + e⁻ˣ) = 5, a quadratic in eˣ after multiplying through by eˣ.

许多双曲方程可通过转换为指数形式求解。例如 2 sinh x + 3 cosh x = 5 化为 (eˣ – e⁻ˣ) + (3/2)(eˣ + e⁻ˣ) = 5，乘以 eˣ 后即得关于 eˣ 的二次方程。

Alternatively, using identities often allows reduction to a single hyperbolic function: if an equation contains only sechx and tanhx, use 1 – tanh²x = sech²x to express everything in terms of tanhx and solve the resulting polynomial.

另一种方法是利用恒等式化为单一双曲函数：若方程只含 sechx 和 tanhx，可利用 1 – tanh²x = sech²x 将所有项用 tanhx 表示，然后解多项式。

Don’t forget inverse definitions when the variable appears both inside and outside the hyperbolic function — logarithmic forms are your key tool.

当变量同时出现在双曲函数内外时，别忘了反函数定义——对数形式是你的关键工具。

10. Hyperbolic Substitutions in Integration | 积分中的双曲代换

For integrals containing √(x²+a²) or √(x²–a²), hyperbolic substitution often works more cleanly than trigonometric substitution. Substitute x = a sinh θ for √(x²+a²) — the radical becomes a cosh θ. For √(x²–a²), use x = a cosh θ, which gives a sinh θ.

对于含有 √(x²+a²) 或 √(x²–a²) 的积分，双曲代换往往比三角代换更简洁。对 √(x²+a²) 用 x = a sinh θ，根号化为 a cosh θ。对 √(x²–a²) 用 x = a cosh θ，根号化为 a sinh θ。

Example: ∫ √(x²+4) dx. Let x = 2 sinh θ, then dx = 2 cosh θ dθ and √(x²+4) = 2 cosh θ. The integral becomes ∫ 4 cosh² θ dθ = 2 ∫ (cosh 2θ + 1) dθ = sinh 2θ +

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Hyperbolic Functions: AQA Exam Essentials | 双曲函数考点精讲