A-Level OCR Mathematics: Hyperbolic Functions Key Points Explained | A-Level OCR 数学：双曲函数考点精讲

📚 A-Level OCR Mathematics: Hyperbolic Functions Key Points Explained | A-Level OCR 数学：双曲函数考点精讲

Hyperbolic functions appear throughout OCR A-Level Further Mathematics – from definitions and graphs to identities, differentiation, integration and solving equations. This article distils the essential knowledge, provides clear comparisons with trigonometric counterparts, and highlights the key techniques examiners expect. Mastering these ideas will equip you to handle any hyperbolic function question with confidence.

双曲函数贯穿 OCR A-Level 进阶数学的多个板块，从定义与图像到恒等式、微分、积分以及方程求解。本文萃取核心知识，给出与三角函数的清晰类比，并突出考官期望的关键方法。吃透这些内容，你就能从容应对任何双曲函数考题。

1. Definition and Origin | 定义与来源

Hyperbolic functions are defined in terms of the exponential function eˣ. The two fundamental functions are the hyperbolic sine and hyperbolic cosine: sinh x = (eˣ – e⁻ˣ)/2 and cosh x = (eˣ + e⁻ˣ)/2. Because they are built from exponentials, they share many algebraic properties with trigonometric functions but with important sign differences.

双曲函数通过指数函数 eˣ 定义。两个基本函数是双曲正弦和双曲余弦：sinh x = (eˣ – e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2。由于从指数构建而来，它们与三角函数有许多相似的代数性质，但存在关键的符号差异。

The names arise from the connection to the hyperbola x² – y² = 1, just as circular functions relate to the unit circle x² + y² = 1. If you set x = cosh t and y = sinh t, then cosh² t – sinh² t = 1, tracing the right-hand branch of the hyperbola. This geometric link explains the ‘hyperbolic’ label.

‘双曲’之名源于它们与双曲线 x² – y² = 1 的联系，正如圆函数对应于单位圆 x² + y² = 1。若令 x = cosh t，y = sinh t，则有 cosh² t – sinh² t = 1，描绘出双曲线的右支。这一几何背景解释了“双曲”的称呼。

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

2. Definitions of tanh, sech, cosech and coth | 其他双曲函数的定义

The remaining hyperbolic functions are defined as ratios, mirroring trigonometric definitions. 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 = cosh x / sinh x. These appear in differentiation, integration and hyperbolic identities.

其余双曲函数由比值定义，与三角函数的定义方式一致。双曲正切为 tanh x = sinh x / cosh x = (eˣ – e⁻ˣ)/(eˣ + e⁻ˣ)。倒数函数分别为 sech x = 1/cosh x，cosech x = 1/sinh x 以及 coth x = 1/tanh x = cosh x / sinh x。它们在微分、积分和双曲恒等式中都会用到。

Unlike cosh x, which is always ≥ 1, tanh x lies strictly between –1 and 1, and coth x has |coth x| > 1 except for a discontinuity at x = 0. Knowing the domains and ranges is vital when solving equations or finding inverse values.

与始终 ≥ 1 的 cosh x 不同，tanh x 严格介于 –1 和 1 之间，coth x 的绝对值大于 1，但在 x = 0 处存在间断点。掌握定义域和值域对求解方程或求反函数值至关重要。

tanh x = sinh x / cosh x, sech x = 1/cosh x, cosech x = 1/sinh x, coth x = cosh x / sinh x

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

The graph of y = sinh x is odd and passes through the origin, growing like ½ eˣ for large positive x and like –½ e⁻ˣ for large negative x. The curve has no asymptotes and is increasing everywhere. Its derivative is cosh x, which is always positive.

y = sinh x 是奇函数，经过原点，当 x 很大时如同 ½ eˣ 增长，当 x 负很大时如同 –½ e⁻ˣ 变化。曲线没有渐近线，且处处递增。其导数为 cosh x，始终为正。

The graph of y = cosh x is symmetric about the y-axis (even) with a minimum point (0, 1). It grows exponentially for both large positive and negative x and is often called a catenary – the shape of a hanging chain. cosh x ≥ 1 for all real x.

y = cosh x 的图像关于 y 轴对称（偶函数），最低点为 (0,1)。当 x 无论正向还是负向很大时均呈指数增长，常被称为悬链线——悬挂链条的形状。对所有实数 x 有 cosh x ≥ 1。

y = tanh x is an odd S-shaped curve with horizontal asymptotes y = 1 and y = –1. It passes through the origin with gradient 1. The graphs of sech x, cosech x and coth x exhibit vertical asymptotes and are useful for sketching composite functions.

y = tanh x 是奇函数，呈 S 形，具有水平渐近线 y = 1 和 y = –1，经过原点且原点处斜率为 1。sech x、cosech x 和 coth x 的图像具有垂直渐近线，在描绘复合函数时很有用。

4. Hyperbolic Identities | 双曲恒等式

The fundamental identity cosh² x – sinh² x = 1 is the hyperbolic counterpart of sin² x + cos² x = 1. From it, two secondary identities follow: 1 – tanh² x = sech² x and coth² x – 1 = cosech² x. The sign pattern differs from the trigonometric versions; these are often the key to simplifying expressions.

基本恒等式 cosh² x – sinh² x = 1 是 sin² x + cos² x = 1 的双曲对应。由此可推出两个次级恒等式：1 – tanh² x = sech² x 和 coth² x – 1 = cosech² x。符号模式与三角版本不同，这些往往是简化表达式的关键。

Addition formulas resemble trigonometric ones with sign changes: sinh(A ± B) = sinhA coshB ± coshA sinhB, cosh(A ± B) = coshA coshB ± sinhA sinhB. Notice the alternating signs in cosh(A – B). The double argument identities are sinh 2x = 2 sinh x cosh x and cosh 2x = cosh² x + sinh² x = 2 cosh² x – 1 = 1 + 2 sinh² x, exactly mirroring the trigonometric forms except for a sign in cosh 2x.

和角公式与三角函数相似但有符号变化：sinh(A ± B)=sinhA coshB ± coshA sinhB，cosh(A ± B)=coshA coshB ± sinhA sinhB。注意 cosh(A – B) 中交替的符号。倍角公式为 sinh 2x = 2 sinh x cosh x 以及 cosh 2x = cosh² x + sinh² x = 2 cosh² x – 1 = 1 + 2 sinh² x，除了 cosh 2x 的一个符号外，完全与三角函数形式一致。

The factor formulae, which express sums and products, also have hyperbolic versions. For instance, sinh A + sinh B = 2 sinh((A+B)/2) cosh((A–B)/2). These are derived directly from the exponential definitions and are useful in integration.

和差化积公式同样有双曲版本。例如 sinh A + sinh B = 2 sinh((A+B)/2) cosh((A–B)/2)。这些均可直接从指数定义推导，并在积分中发挥作用。

5. Osborn’s Rule | 奥斯本规则

Osborn’s rule provides a quick way to convert any trigonometric identity into its hyperbolic counterpart. Replace each circular function directly: sin → sinh, cos → cosh, tan → tanh, etc. Then change the sign of every term that contains a product of two sines, implicitly or explicitly. In practice, this means a product such as sin² x becomes –sinh² x in the hyperbolic version.

奥斯本规则提供了一种将任何三角恒等式快速转换为双曲恒等式的方法。直接将圆函数替换：sin → sinh，cos → cosh，tan → tanh 等。然后改变每一个显式或隐式含有两个正弦乘积的项的符号。实践中，这意味着 sin² x 这种乘积在双曲版本中变为 –sinh² x。

For example, starting from cos 2θ = 1 – 2 sin² θ, replacing cos → cosh and sin → sinh gives cosh 2θ = 1 – 2 sinh² θ; the product sin² θ contains two sines, so its sign flips relative to the trigonometric identity, making it +? Wait, trigonometric is cos 2θ = 1 – 2 sin² θ. Replace sin with sinh, product sin² θ is treated as a product of two sines → –sinh² θ. So cosh 2θ = 1 – 2(–sinh² θ) = 1 + 2 sinh² θ, which indeed matches the known identity. The rule always works and saves memorising every possible identity.

例如，从 cos 2θ = 1 – 2 sin² θ 出发，替换 cos → cosh、sin → sinh 得到 cosh 2θ = 1 – 2 sinh² θ；但 sin² θ 包含两个正弦的乘积，因此其符号相对于三角恒等式要翻转。所以在双曲版本中该项变为 –(–2 sinh² θ) = +2 sinh² θ，最终 cosh 2θ = 1 + 2 sinh² θ，与已知恒等式一致。此规则屡试不爽，可省去记忆每一个可能的恒等式。

6. Inverse Hyperbolic Functions | 反双曲函数

The inverse hyperbolic functions are denoted arsinh x, arcosh x and artanh x (some textbooks use sinh⁻¹ x, etc.). Their domains and ranges must be known: arsinh x is defined for all real x; arcosh x is defined for x ≥ 1 and returns values ≥ 0; artanh x has domain |x| < 1 and range all real numbers. The graphs of these inverses can be obtained by reflecting the respective restricted functions in y = x.

反双曲函数记作 arsinh x、arcosh x 和 artanh x（部分教材使用 sinh⁻¹ x 等）。务必熟记其定义域与值域：arsinh x 对所有实数 x 定义；arcosh x 要求 x ≥ 1，且返回值 ≥ 0；artanh x 的定义域为 |x| < 1，值域为全体实数。这些反函数的图像可由限制后的原函数关于 y = x 反射得到。

Differentiating inverse hyperbolic functions yields standard results that appear frequently in exams. The derivatives are: d/dx (arsinh x) = 1/√(1 + x²), d/dx (arcosh x) = 1/√(x² – 1), d/dx (artanh x) = 1/(1 – x²) for |x| < 1. These can be proved using implicit differentiation or the logarithmic forms.

对反双曲函数求导将得到考试中频繁出现的标准结果。导数为：d/dx (arsinh x) = 1/√(1 + x²)，d/dx (arcosh x) = 1/√(x² – 1)，d/dx (artanh x) = 1/(1 – x²)（|x| < 1）。这些可借助隐函数求导或对数形式加以证明。

7. Logarithmic Equivalents | 对数等价形式

Because hyperbolic functions are expressed via exponentials, their inverses have logarithmic forms. These are very useful for evaluating exact values or solving equations algebraically: arsinh x = ln(x + √(x² + 1)), arcosh x = ln(x + √(x² – 1)) for x ≥ 1, and artanh x = ½ ln((1 + x)/(1 – x)) for |x| < 1. OCR expects you to be able to derive these by setting y = arsinh x, rewriting into exponentials and solving for eʸ.

由于双曲函数由指数表示，其反函数具有对数形式。这在求精确值或代数求解方程时非常有用：arsinh x = ln(x + √(x² + 1))，arcosh x = ln(x + √(x² – 1))（x ≥ 1），artanh x = ½ ln((1 + x)/(1 – x))（|x| < 1）。OCR 考试期望你能通过设 y = arsinh x、化为指数方程并求解 eʸ 来推导这些式子。

Armed with the logarithmic forms, you can solve equations such as cosh x = 2 exactly without a calculator: x = arcosh 2 = ln(2 + √3). Similarly, equations that mix exponentials and hyperbolics can be tackled by converting everything to exponentials and using a substitution such as u = eˣ.

掌握对数形式后，便可精确求解诸如 cosh x = 2 的方程而无需计算器：x = arcosh 2 = ln(2 + √3)。同样，那些混合了指数与双曲函数的方程可通过全部化为指数形式并使用替换 u = eˣ 来处理。

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

The derivatives of the basic hyperbolic functions are cyclical and should be memorised: d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x (no negative sign, unlike trigonometric cos), d/dx (tanh x) = sech² x. The derivatives of the reciprocal functions are d/dx (coth x) = –cosech² x, d/dx (sech x) = –sech x tanh x, d/dx (cosech x) = –cosech x coth x. Note the negative signs, which often catch students out.

基本双曲函数的导数具有循环性，务必熟记：d/dx (sinh x) = cosh x，d/dx (cosh x) = sinh x（与三角函数不同，没有负号），d/dx (tanh x) = sech² x。倒数函数的导数为：d/dx (coth x) = –cosech² x，d/dx (sech x) = –sech x tanh x，d/dx (cosech x) = –cosech x coth x。请注意那些负号，常使考生失分。

When the argument is a linear function of x, the chain rule applies in the usual way. For example, d/dx (sinh(ax + b)) = a cosh(ax + b). For more complicated arguments, careful systematic differentiation is required. Be ready to combine these derivatives with product, quotient and chain rules.

当变量为 x 的线性函数时，链式法则照常使用。例如 d/dx (sinh(ax + b)) = a cosh(ax + b)。对于更复杂的变量，需要系统且仔细地求导。请准备好将这些导数与乘法法则、除法法则和链式法则结合使用。

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

Standard integrals are the reverse of the derivatives: ∫ sinh x dx = cosh x + C, ∫ cosh x dx = sinh x + C, ∫ sech² x dx = tanh x + C, ∫ cosech² x dx = –coth x + C, ∫ sech x tanh x dx = –sech x + C, ∫ cosech x coth x dx = –cosech x + C. These results must be at your fingertips for both direct integration and use within substitution methods.

标准积分是导数的逆运算：∫ sinh x dx = cosh x + C，∫ cosh x dx = sinh x + C，∫ sech² x dx = tanh x + C，∫ cosech² x dx = –coth x + C，∫ sech x tanh x dx = –sech x + C，∫ cosech x coth x dx = –cosech x + C。无论是直接积分还是用于代换法，这些结果都必须信手拈来。

A very powerful technique is integration by substitution using hyperbolic functions. Integrals involving √(a² + x²) can be simplified with x = a sinh θ; those with √(x² – a²) use x = a cosh θ; and √(a² – x²) appears less often but can use x = a tanh θ or remain trigonometric. The key is recognising the form and confidently swapping variables.

一个非常有力的技巧是使用双曲函数作代换积分。含有 √(a² + x²) 的积分可用 x = a sinh θ 化简；含有 √(x² – a²) 的用 x = a cosh θ；虽然 √(a² – x²) 不常见，但可使用 x = a tanh θ 或仍用三角代换。关键在于识别形式并自信地进行变量代换。

When integrating rational functions of exponentials, rewriting sinh and cosh in terms of eˣ and setting u = eˣ can turn the integral into a standard rational function. Also, integrals of tanh x and coth x follow from logarithmic integration: ∫ tanh x dx = ln(cosh x) + C and ∫ coth x dx = ln|sinh x| + C.

积分指数函数的有理式时，可将 sinh 和 cosh 写成 eˣ 的形式并设 u = eˣ，从而化作标准有理函数积分。此外，tanh x 和 coth x 的积分由对数积分给出：∫ tanh x dx = ln(cosh x) + C，∫ coth x dx = ln|sinh x| + C。

10. Solving Equations with Hyperbolic Functions | 双曲函数方程求解

Equations such as 5 cosh x – 3 sinh x = 4 are best solved by converting to exponentials: replace cosh x and sinh x with their definitions, multiply through by eˣ to obtain a quadratic in eˣ, solve for eˣ, and finally take natural logarithms. This exponential method is robust and works for many linear combinations.

诸如 5 cosh x – 3 sinh x = 4 的方程，最佳解法是化为指数形式：用定义替换 cosh x 和 sinh x，两边同乘 eˣ 得到关于 eˣ 的二次方程，解出 eˣ 然后取自然对数。这种指数方法十分可靠，适用于许多线性组合。

If an equation involves a single hyperbolic function, you can often use identities to simplify. For instance, 2 cosh² x + sinh x = 2 can be turned into a quadratic in sinh x using cosh² x = 1 + sinh² x. Remember to check solutions against the domains of any inverse or logarithmic forms used later.

若方程仅涉及单一双曲函数，通常可利用恒等式化简。例如，2 cosh² x + sinh x = 2 可通过 cosh² x = 1 + sinh² x 化为关于 sinh x 的二次方程。注意根据后续使用的反函数或对数形式的定义域来检验解。

Equations involving inverse hyperbolic functions are less common but may appear. Use the logarithmic definitions to rewrite the equation algebraically. For example, arsinh x = ln 2 leads to x + √(x² + 1) = 2, which can be solved by squaring carefully. Always verify the solution satisfies the original domain restrictions.

涉及反双曲函数的方程虽不常见，但仍可能出现。利用对数定义将方程改写为代数形式。例如 arsinh x = ln 2 导出 x + √(x² + 1) = 2，通过小心平方可求解。务必验证解满足原定义域限制。

11. Graphing and Transformations | 图像变换与分析

Exam questions often ask you to sketch a transformed hyperbolic function, such as y = 3 cosh(2x – 1) – 4, or to find its range and any stationary points. Use the standard graph of cosh x as a base, apply the appropriate stretches, translations and reflections, and mark key features like the minimum point, axis of symmetry and asymptotes if present.

试题常要求你描绘一个变换后的双曲函数图像，比如 y = 3 cosh(2x – 1) – 4，或求其值域和驻点。以 cosh x 的标准图像为基础，施加适当的伸缩、平移和反射，并标出关键特征，如最小值点、对称轴以及可能存在的渐近线。

For curves like y = artanh x + 1, note that the domain is unchanged but the range shifts upward by 1, causing the horizontal asymptotes to move from y = –∞/∞? Actually artanh x has range (−∞, ∞), so no horizontal asymptotes in the range, but the graph has vertical asymptotes at x = ±1. Transformations of inverse hyperbolic functions follow the usual rules; just be careful with restricted domains.

对于 y = artanh x + 1 这样的曲线，需注意定义域不变而值域向上平移 1，导致渐近线移动；artanh x 本身值域为 (−∞, ∞)，因此没有水平渐近线，但在 x = ±1 处有垂直渐近线。反双曲函数的变换遵循常规法则，只要细心处理受限制的定义域即可。

12. Exam Technique and Common Pitfalls | 应试技巧与常见误区

In an OCR A-Level paper, hyperbolic function questions can appear in pure sections, often linking differentiation, integration and differential equations. When you see √(1 + x²) or √(x² – 1) in an integral, immediately think of a hyperbolic substitution. Know your standard integrals and derivatives by heart; many students lose time trying to re-derive them under pressure.

在 OCR A-Level 试卷中，双曲函数问题可出现在纯数部分，常联系微分、积分和微分方程。当积分中出现 √(1 + x²) 或 √(x² – 1) 时，立刻想到双曲代换。将标准积分和导数熟记于心；许多学生因临场重新推导而浪费时间。

A common mistake is mismanaging signs when applying Osborn’s rule or differentiating. For example, the derivative of cosh x is sinh x, not –sinh x. For integration, ∫ sinh x dx = cosh x + C, whereas ∫ cos x dx = sin x + C; don’t confuse the two. Also check that an expression like artanh x is only valid for |x| < 1, and adjust if the question gives values outside this range.

常见错误是在应用奥斯本规则或求导时将符号弄混。例如，cosh x 的导数是 sinh x，而非 –sinh x。积分时 ∫ sinh x dx = cosh x + C，而 ∫ cos x dx = sin x + C，切莫混淆两者。还要检查类似 artanh x 的表达式只在 |x| < 1 时有效，若题目给出的值超出此范围则需调整。

Lastly, practice deriving the logarithmic forms for the inverse functions and using them to solve equations exactly. These derivations are worth marks and give you a deeper understanding. Consistent practice with past paper questions, especially those that combine hyperbolic functions with integration by parts or differential equations, will build confidence.

最后，练习推导反函数的对数形式并用它们精确求解方程。这些推导过程本身即可得分，并加深理解。反复演练往年真题，尤其是那些将双曲函数与分部积分或微分方程结合的题目，能帮助你建立信心。

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A-Level OCR Mathematics: Hyperbolic Functions Key Points Explained | A-Level OCR 数学：双曲函数 考点精讲