Edexcel Maths: Hyperbolic Functions | 双曲函数考点精讲

📚 Edexcel Maths: Hyperbolic Functions | 双曲函数考点精讲

Hyperbolic functions are an essential topic in Edexcel A-Level Further Mathematics. They appear frequently in Pure Mathematics exams, particularly in calculus, identities, and equation-solving. This comprehensive revision guide covers all the key exam points: definitions, graphs, identities, differentiation, integration, inverse functions, and common pitfalls. Mastering these concepts will boost your confidence and help you secure high marks.

双曲函数是Edexcel A-Level进阶数学的重要考点。它们在纯数学考试中频繁出现，尤其涉及微积分、恒等式和方程求解。本篇考点精讲全面梳理了所有关键内容：定义、图像、恒等式、微分、积分、反函数以及常见易错点。掌握这些概念将增强你的信心，帮助你斩获高分。

1. Definitions and Notation | 定义与记法

The hyperbolic sine and cosine are defined using exponential functions: sinh x = (eˣ – e⁻ˣ)/2 and cosh x = (eˣ + e⁻ˣ)/2. From these, the hyperbolic tangent is tanh x = sinh x / cosh x = (eˣ – e⁻ˣ)/(eˣ + e⁻ˣ). The reciprocal functions are sech x = 1/cosh x, cosech x = 1/sinh x, and coth x = 1/tanh x. It is crucial to memorise these exponential forms as they are the foundation for all identities and calculus.

双曲正弦和双曲余弦是通过指数函数定义的：sinh x = (eˣ – e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2。由此可得双曲正切 tanh x = sinh x / cosh x = (eˣ – e⁻ˣ)/(eˣ + e⁻ˣ)。倒易函数为 sech x = 1/cosh x，cosech x = 1/sinh x，coth x = 1/tanh x。熟记这些指数形式至关重要，因为它们是所有恒等式和微积分的基础。

2. Graphs and Properties | 图像与性质

You must be able to sketch y = sinh x, y = cosh x, and y = tanh x. sinh x is an odd function, passing through the origin with a shape similar to a cubic but steeper. cosh x is an even function, always ≥1, with a minimum point at (0,1) and exponential growth for large |x|. tanh x is odd, with horizontal asymptotes at y = ±1. Understand the domains and ranges: for sinh, domain and range are all real numbers; for cosh, domain is ℝ, range is [1,∞); for tanh, domain is ℝ, range is (-1,1).

你必须能画出 y = sinh x, y = cosh x 和 y = tanh x 的草图。sinh x 是奇函数，通过原点，形状类似三次函数但更陡峭。cosh x 是偶函数，恒 ≥1，极小值点 (0,1)，当 |x| 很大时呈指数增长。tanh x 也是奇函数，有水平渐近线 y = ±1。要理解定义域和值域：sinh 的定义域和值域均为全体实数；cosh 的定义域为 ℝ，值域为 [1,∞)；tanh 的定义域为 ℝ，值域为 (-1,1)。

3. Basic Identities and Osborne’s Rule | 基本恒等式与奥斯本法则

The fundamental identity is cosh² x – sinh² x = 1. This mirrors the trigonometric identity cos² x + sin² x = 1, but with a sign change. Osborne’s rule states that any trigonometric identity involving sine or cosine can be converted into a hyperbolic identity by replacing cos with cosh, sin with i sinh, and changing the sign of any term containing a product of two sines (or implicitly using i² = -1). For example, from cos 2x = 1 – 2 sin² x we get cosh 2x = 1 + 2 sinh² x. Other key identities: cosh 2x = cosh² x + sinh² x = 2 cosh² x – 1 = 2 sinh² x + 1; sinh 2x = 2 sinh x cosh x; tanh² x + sech² x = 1; coth² x – cosech² x = 1.

基本恒等式是 cosh² x – sinh² x = 1。它类似于三角恒等式 cos² x + sin² x = 1，但符号有变化。奥斯本法则指出，任何包含正弦或余弦的三角恒等式都可以转换为双曲恒等式，只需将 cos 换成 cosh，sin 换成 i sinh，并改变含有两个正弦乘积的项的符号（即隐含使用了 i² = -1）。例如，从 cos 2x = 1 – 2 sin² x 可得到 cosh 2x = 1 + 2 sinh² x。其他关键恒等式：cosh 2x = cosh² x + sinh² x = 2 cosh² x – 1 = 2 sinh² x + 1；sinh 2x = 2 sinh x cosh x；tanh² x + sech² x = 1；coth² x – cosech² x = 1。

4. Inverse Hyperbolic Functions | 反双曲函数

The inverse hyperbolic functions are arsinh x, arcosh x, and artanh x (sometimes written as sinh⁻¹ etc.). It is essential to know their logarithmic forms, which are derived from the exponential definitions. Edexcel expects you to derive or quote: arsinh x = ln(x + √(x²+1)), valid for all real x. arcosh x = ln(x + √(x²-1)), with domain x ≥ 1 and principal value ≥ 0. artanh x = (1/2) ln((1+x)/(1-x)), for |x| < 1. You may be asked to prove these in the exam.

反双曲函数包括 arsinh x, arcosh x 和 artanh x（有时记作 sinh⁻¹ 等）。必须掌握它们的对数形式，这从指数定义推导而来。Edexcel 考试要求你推导或直接引用：arsinh x = ln(x + √(x²+1))，对所有实数 x 有效。arcosh x = ln(x + √(x²-1))，定义域为 x ≥ 1，主值 ≥ 0。artanh x = (1/2) ln((1+x)/(1-x))，其中 |x| < 1。考试中可能会要求你证明这些公式。

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

Differentiation of hyperbolic functions is straightforward once you know the exponential forms or the defining relationships. The key derivatives are: d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech² x. For reciprocal functions: d/dx (cosech x) = -cosech x coth x, d/dx (sech x) = -sech x tanh x, d/dx (coth x) = -cosech² x. Be careful with the sign differences compared to trigonometric derivatives. Chain rule applies as usual.

双曲函数的微分一旦掌握了指数形式或基本关系就很简单。关键导数是：d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech² x。对于倒易函数：d/dx (cosech x) = -cosech x coth x, d/dx (sech x) = -sech x tanh x, d/dx (coth x) = -cosech² x。注意与三角函数的导数在符号上的差异。链式法则照常使用。

For inverse hyperbolic functions: 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 required for integration as well.

对于反双曲函数：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。这些也是积分所需。

6. Integrals Leading to Hyperbolic Functions | 导出双曲函数的积分

Be familiar with standard integrals: ∫ sinh x dx = cosh x + C, ∫ cosh x dx = sinh x + C, ∫ tanh x dx = ln|cosh x| + C. More importantly, recognise forms that integrate to inverse hyperbolic functions: ∫ dx/√(x²+1) = arsinh x + C, ∫ dx/√(x²-1) = arcosh x + C (for x>1), and ∫ dx/(1-x²) = artanh x + C (for |x|<1) or arcoth x depending on domain. These are exam favourites, often requiring completion of the square or substitution first.

要熟悉标准积分：∫ sinh x dx = cosh x + C, ∫ cosh x dx = sinh x + C, ∫ tanh x dx = ln|cosh x| + C。更重要的是识别可积分为反双曲函数的类型：∫ dx/√(x²+1) = arsinh x + C, ∫ dx/√(x²-1) = arcosh x + C (x>1), ∫ dx/(1-x²) = artanh x + C (当 |x|<1) 或根据定义域为 arcoth x。这些是考试热门题型，通常需要先完成配平方或代换。

7. Solving Hyperbolic Equations | 解双曲方程

Examiners often set equations like a sinh x + b cosh x = c, where you need to substitute the exponential definitions and solve a quadratic in eˣ. Alternatively, use identities to reduce an equation to a single type of function. For example, solve cosh 2x = 5 sinh x + 1 by using the double-angle identity cosh 2x = 1 + 2 sinh² x. Then you get 2 sinh² x – 5 sinh x = 0, which factors to sinh x (2 sinh x – 5) = 0. Solve for sinh x and then

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