Hyperbolic Identities | 双曲函数恒等式

📚 Hyperbolic Identities | 双曲函数恒等式

Hyperbolic functions appear in many areas of mathematics, from calculus to differential equations. Understanding their identities is essential for AQA A-level Mathematics, particularly when solving equations and integrating certain functions. This revision guide provides a complete, exam-focused overview of hyperbolic identities, proofs, and typical questions.

双曲函数出现在数学的许多领域，从微积分到微分方程。理解它们的恒等式对于 AQA A-level 数学至关重要，尤其是在解方程和积分特定函数时。本复习指南提供关于双曲恒等式、证明和典型题型的完整、紧扣考点的概述。

1. Definitions and Exponential Forms | 定义与指数形式

The hyperbolic sine and cosine are defined using the exponential functions:

双曲正弦和双曲余弦通过指数函数定义如下：

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

From these, the other hyperbolic functions are defined as tanh x = sinh x / cosh x, coth x = cosh x / sinh x, sech x = 1 / cosh x, and csch x = 1 / sinh x.

由此，其他双曲函数定义为：tanh x = sinh x / cosh x，coth x = cosh x / sinh x，sech x = 1 / cosh x，csch x = 1 / sinh x。

You must be able to recall these exponential forms and use them to derive identities from first principles.

你必须能够记住这些指数形式，并用它们从第一性原理推导恒等式。

2. Pythagorean Identity | 毕达哥拉斯恒等式

The most fundamental hyperbolic identity is the analogue of sin²x + cos²x = 1:

最基本的双曲恒等式是 sin²x + cos²x = 1 的类比：

cosh²x − sinh²x = 1

Dividing by cosh²x gives 1 − tanh²x = sech²x, while dividing by sinh²x gives coth²x − 1 = csch²x.

两边除以 cosh²x 得 1 − tanh²x = sech²x；两边除以 sinh²x 得 coth²x − 1 = csch²x。

These forms are frequently used to simplify expressions and prove other identities.

这些形式常用于化简表达式和证明其他恒等式。

3. Addition and Subtraction Formulas | 加法与减法公式

The addition formulas for hyperbolic functions closely resemble those for trigonometric functions, but note the sign in cosh(x − y).

双曲函数的加法公式与三角函数非常相似，但要注意 cosh(x − y) 中的符号。

sinh(x ± y) = sinh x cosh y ± cosh x sinh y

cosh(x ± y) = cosh x cosh y ± sinh x sinh y

For tanh:

对于 tanh：

tanh(x ± y) = (tanh x ± tanh y) / (1 ± tanh x tanh y)

Notice that the formula for cosh(x − y) contains a plus sign, unlike cos(x − y) which has a minus sign.

注意 cosh(x − y) 的公式中是加号，不像 cos(x − y) 中是减号。

4. Double-Angle and Half-Angle Identities | 二倍角与半角恒等式

Setting y = x in the addition formulas gives the double-angle identities:

在加法公式中令 y = x 可得二倍角恒等式：

sinh 2x = 2 sinh x cosh x

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

tanh 2x = 2 tanh x / (1 + tanh²x)

The half-angle versions follow directly:

半角版本直接推出：

cosh²(x/2) = (cosh x + 1)/2, sinh²(x/2) = (cosh x − 1)/2

These are useful when integrating or solving equations involving 2x or x/2.

这些在积分或解包含 2x 或 x/2 的方程时非常有用。

5. Reciprocal and Quotient Identities | 倒数与商数恒等式

The quotient and reciprocal relationships are essential for rewriting expressions:

商和倒数关系对于改写表达式至关重要：

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

sech x = 1 / cosh x, csch x = 1 / sinh x

For example, you may use sech²x = 1 − tanh²x to replace sec²x = 1 + tan²x in trigonometric contexts.

例如，可以用 sech²x = 1 − tanh²x 来替代三角函数情形中的 sec²x = 1 + tan²x。

Always check that the domain of each function is respected when using these identities.

使用这些恒等式时，要始终注意每个函数的定义域。

6. Relationship with Trigonometric Identities (Osborn’s Rule) | 与三角恒等式的关系（奥斯本规则）

Most trigonometric identities have a hyperbolic analogue. Osborn’s rule states: replace sin by sinh and cos by cosh; then change the sign of any term that contains a product of two sine functions.

大多数三角恒等式都有双曲类比。奥斯本规则指出：将 sin 替换为 sinh，cos 替换为 cosh；然后改变任何包含两个正弦函数乘积的项的符号。

For example, cos 2x = cos²x − sin²x becomes cosh 2x = cosh²x + sinh²x, because the term sin²x changes sign.

例如，cos 2x = cos²x − sin²x 变成 cosh 2x = cosh²x + sinh²x，因为 sin²x 的符号改变了。

Similarly, sin(x + y) = sin x cos y + cos x sin y becomes sinh(x + y) = sinh x cosh y + cosh x sinh y with no sign change, since there is only one product of sin and cos.

类似地，sin(x + y) = sin x cos y + cos x sin y 变成 sinh(x + y) = sinh x cosh y + cosh x sinh y，因为没有出现“两个正弦”的乘积项，所以符号不变。

7. Proving Hyperbolic Identities | 证明双曲恒等式

The most reliable method for proving a hyperbolic identity is to use the exponential definitions.

证明双曲恒等式最可靠的方法是使用指数定义。

Example: Prove cosh²x − sinh²x = 1.

例：证明 cosh²x − sinh²x = 1。

cosh²x − sinh²x = [(eˣ + e⁻ˣ)/2]² − [(eˣ − e⁻ˣ)/2]²

Expanding both squares gives (e²ˣ + 2 + e⁻²ˣ − e²ˣ + 2 − e⁻²ˣ)/4 = 4/4 = 1.

展开两个平方得 (e²ˣ + 2 + e⁻²ˣ − e²ˣ + 2 − e⁻²ˣ)/4 = 4/4 = 1。

Always write down each step clearly in exam proofs, as method marks are awarded for correct substitution.

在考试证明中，每一步都要清楚写下，因为正确代入会获得方法分。

8. Derivatives and Related Identities | 导数与相关恒等式

The derivatives of hyperbolic functions form a set of identities that are frequently tested:

双曲函数的导数构成一组常考恒等式：

d/dx sinh x = cosh x, d/dx cosh x = sinh x

d/dx tanh x = sech²x, d/dx coth x = −csch²x

d/dx sech x = −sech x tanh x, d/dx csch x = −csch x coth x

These are crucial for integration: for example, ∫ sech²x dx = tanh x + C.

这些对积分至关重要：例如，∫ sech²x dx = tanh x + C。

Note the negative signs in the derivatives of coth, sech and csch.

注意 coth、sech 和 csch 的导数中的负号。

9. Inverse Hyperbolic Functions and Logarithmic Identities | 反双曲函数与对数恒等式

Inverse hyperbolic functions can be written in logarithmic form, which is useful for solving equations.

反双曲函数可以写成对数形式，这在解方程时很有用。

arsinh x = ln(x + √(x² + 1))

arcosh x = ln(x + √(x² − 1)), x ≥ 1

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

For example, solving cosh x = 3 gives x = ± arcosh 3 = ± ln(3 + 2√2).

例如，解 cosh x = 3 得 x = ± arcosh 3 = ± ln(3 + 2√2)。

Check domain restrictions carefully, especially for arcosh and artanh.

要仔细检查定义域限制，尤其是 arcosh 和 artanh。

10. Solving Equations Using Identities | 用恒等式解方程

Hyperbolic identities can reduce complicated equations to simpler quadratic or logarithmic forms.

双曲恒等式可以将复杂方程转化为更简单的二次或对数形式。

Example: Solve 2 cosh x − sinh x = 2 for real x.

例：解 2 cosh x − sinh x = 2，其中 x 为实数。

Using the exponential definitions, 2·(eˣ + e⁻ˣ)/2 − (eˣ − e⁻ˣ)/2 = 2, which simplifies to eˣ + 3e⁻ˣ = 4.

使用指数定义，2·(eˣ + e⁻ˣ)/2 − (eˣ − e⁻ˣ)/2 = 2，化简得 eˣ + 3e⁻ˣ = 4。

Multiplying by eˣ gives e²ˣ − 4eˣ + 3 = 0, so eˣ = 1 or 3, hence x = 0 or ln 3.

乘以 eˣ 得 e²ˣ − 4eˣ + 3 = 0，所以 eˣ = 1 或 3，因此 x = 0 或 ln 3。

Always verify that your solutions lie in the valid domain of the original equation.

始终验证你的解是否在原方程的有效定义域内。

11. Common Mistakes and Exam Tips | 常见错误与考试提示

Do not confuse hyperbolic identities with trigonometric ones; for example, cosh²x − sinh²x = 1, not sin²x + cos²x = 1.

不要将双曲恒等式与三角恒等式混淆；例如，cosh²x − sinh²x = 1，而不是 sin²x + cos²x = 1。
Remember the sign change in Osborn’s rule; forgetting it is a frequent error in double-angle formulas.

记住奥斯本规则中的符号变化；忘记它是二倍角公式中常见的错误。
When proving identities, write down the exponential form first and show algebraic simplification fully.

证明恒等式时，先写出指数形式，并完整展示代数化简过程。
For equations, always check for extraneous roots after multiplying by eˣ or using logarithms.

对于方程，在乘以 eˣ 或使用对数后，要检查是否有增根。

12. Summary | 总结

The key hyperbolic identities to remember are: cosh²x − sinh²x = 1; sinh(x ± y) = sinh x cosh y ± cosh x sinh y; cosh(x ± y) = cosh x cosh y ± sinh x sinh y; and the double-angle formulas.

需要记住的关键双曲恒等式有：cosh²x − sinh²x = 1；sinh(x ± y) = sinh x cosh y ± cosh x sinh y；cosh(x ± y) = cosh x cosh y ± sinh x sinh y；以及二倍角公式。

Master the exponential definitions, apply Osborn’s rule cautiously, and practise solving past-paper questions to build confidence.

掌握指数定义，谨慎应用奥斯本规则，并通过练习历年真题来建立信心。

With these tools, you can handle any hyperbolic identity question in your AQA A-level Mathematics exam.

有了这些工具，你就能在 AQA A-level 数学考试中处理任何双曲恒等式问题。

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Hyperbolic Identities | 双曲函数恒等式