Hyperbolic Identities | 双曲恒等式

📚 Hyperbolic Identities | 双曲恒等式

Hyperbolic functions are closely related to the exponential function and share many structural similarities with trigonometric functions. Their identities are essential for simplifying expressions, solving equations, and evaluating integrals in A-Level Further Mathematics.

双曲函数与指数函数密切相关，并且在结构上与三角函数有许多相似之处。双曲恒等式是化简表达式、解方程以及计算积分的重要工具，是 A-Level 进阶数学的核心内容之一。

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

The hyperbolic functions are defined in terms of the exponential functions exp(x) and exp(-x). These definitions are the foundation for all hyperbolic identities.

双曲函数由指数函数 exp(x) 和 exp(-x) 定义。这些定义是所有双曲恒等式的基础。

sinh x = (exp(x) − exp(−x)) / 2, cosh x = (exp(x) + exp(−x)) / 2

From these two primary definitions, the remaining four hyperbolic functions are defined as ratios and reciprocals.

由这两个主要定义，其余四个双曲函数被定义为比值和倒数。

tanh x = sinh x / cosh x = (exp(x) − exp(−x)) / (exp(x) + exp(−x))

tanh x = sinh x / cosh x = (exp(x) − exp(−x)) / (exp(x) + exp(−x))
coth x = 1 / tanh x = cosh x / sinh x, for x ≠ 0

coth x = 1 / tanh x = cosh x / sinh x，其中 x ≠ 0
sech x = 1 / cosh x

sech x = 1 / cosh x
csch x = 1 / sinh x, for x ≠ 0

csch x = 1 / sinh x，其中 x ≠ 0

Unlike trigonometric functions, cosh x is an even function, sinh x and tanh x are odd functions. This parity will affect identities involving signs.

与三角函数不同，cosh x 是偶函数，sinh x 和 tanh x 是奇函数。这种奇偶性会影响涉及符号的恒等式。

2. The Fundamental Identity | 基本恒等式

The most important hyperbolic identity is analogous to the Pythagorean identity in trigonometry. It is derived directly from the exponential definitions.

最重要的双曲恒等式与三角学中的毕达哥拉斯恒等式类似，可以直接由指数定义推导出来。

cosh² x − sinh² x = 1

To verify this, expand using the definitions: cosh² x − sinh² x = ((exp(x)+exp(−x))/2)² − ((exp(x)−exp(−x))/2)² = 1.

为了验证，将定义代入展开：cosh² x − sinh² x = ((exp(x)+exp(−x))/2)² − ((exp(x)−exp(−x))/2)² = 1。

This identity is always true for every real number x. It also implies that the point (cosh t, sinh t) lies on the hyperbola x² − y² = 1, which explains the name “hyperbolic”.

该恒等式对所有实数 x 都成立。它还意味着点 (cosh t, sinh t) 位于双曲线 x² − y² = 1 上，这正是“双曲”名称的来源。

3. Related Pythagorean Identities | 相关的毕达哥拉斯恒等式

Dividing the fundamental identity by cosh² x or sinh² x gives two additional identities that are frequently used in integration and differentiation.

将基本恒等式分别除以 cosh² x 和 sinh² x，可得到两个额外恒等式，常用于积分和求导。

1 − tanh² x = sech² x

coth² x − 1 = csch² x

These identities are structurally identical to the trigonometric versions, except for the sign of the squared term. Osborne’s rule states that when converting a trigonometric identity to a hyperbolic identity, replace sin² with −sinh².

这些恒等式在结构上与三角版本一致，只是平方项的符号不同。奥斯本规则指出：将三角恒等式转换为双曲恒等式时，应将 sin² 替换为 −sinh²。

For example, from 1 + tan² x = sec² x, using the rule gives 1 − tanh² x = sech² x.

例如，由 1 + tan² x = sec² x，利用该规则得到 1 − tanh² x = sech² x。

4. Addition and Subtraction Formulae | 和角与差角公式

The addition formulae for hyperbolic functions mirror the trigonometric addition formulae, with subtle sign changes in the sinh terms.

双曲函数的和角公式与三角函数的和角公式相似，但 sinh 项的符号略有不同。

sinh(x + y) = sinh x cosh y + cosh x sinh y

cosh(x + y) = cosh x cosh y + sinh x sinh y

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

Notice that cosh(x + y) uses a plus sign, whereas cos(x + y) uses a minus sign. This is the key difference to remember.

注意 cosh(x + y) 使用加号，而 cos(x + y) 使用减号。这是需要记住的关键区别。

Replacing y with −y gives the subtraction formulae. Because sinh is odd and cosh is even, we obtain:

将 y 替换为 −y 可得到差角公式。由于 sinh 是奇函数、cosh 是偶函数，我们得到：

sinh(x − y) = sinh x cosh y − cosh x sinh y

cosh(x − y) = cosh x cosh y − sinh x sinh y

5. Double and Half Angle Formulae | 二倍角与半角公式

Setting y = x in the addition formulae produces the double angle formulae, which are essential for solving equations and simplifying integrals.

在和角公式中令 y = x，即可得到二倍角公式，这在解方程和化简积分中非常重要。

sinh 2x = 2 sinh x cosh x

cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x

The three equivalent forms for cosh 2x are especially useful: the first is symmetric, the second allows converting cosh² to cosh 2x, and the third converts sinh² to cosh 2x.

cosh 2x 的三种等价形式特别有用：第一种是对称形式，第二种可将 cosh² 转换为 cosh 2x，第三种可将 sinh² 转换为 cosh 2x。

Solving for cosh² x and sinh² x gives the half angle (power-reduction) identities:

解出 cosh² x 和 sinh² x，可得到半角（降幂）恒等式：

cosh² x = (cosh 2x + 1) / 2

sinh² x = (cosh 2x − 1) / 2

These are often used when integrating expressions involving squares of hyperbolic functions.

这些公式常用于对含有双曲函数平方的表达式进行积分。

6. Inverse Hyperbolic Functions | 反双曲函数

The inverse hyperbolic functions are denoted arsinh, arcosh and artanh. They can be expressed in logarithmic form, which is essential for solving equations and evaluating integrals.

反双曲函数记作 arsinh、arcosh 和 artanh。它们都可以用对数形式表示，这对解方程和计算积分至关重要。

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 logarithmic forms are derived by solving for exp(y) in the original definitions and then taking logarithms.

通过对原定义中解出 exp(y) 再取对数，即可推导出这些对数形式。

For example, if sinh y = x, then exp(y) − exp(−y) = 2x. Multiplying by exp(y) yields a quadratic in exp(y), giving exp(y) = x + √(x² + 1).

例如，若 sinh y = x，则 exp(y) − exp(−y) = 2x。两边乘以 exp(y) 得到关于 exp(y) 的二次方程，解得 exp(y) = x + √(x² + 1)。

7. Derivatives and Integrals | 导数与积分

The derivatives of hyperbolic functions follow patterns similar to trigonometric functions, but with some important sign differences.

双曲函数的导数与三角函数类似，但存在重要的符号差异。

Function	Derivative	函数	导数
sinh x	cosh x	sinh x	cosh x
cosh x	sinh x	cosh x	sinh x
tanh x	sech² x	tanh x	sech² x
coth x	−csch² x	coth x	−csch² x
sech x	−sech x tanh x	sech x	−sech x tanh x
csch x	−csch x coth x	csch x	−csch x coth x

Corresponding integral results are often used in A-Level questions, especially those involving 1 / √(x² + 1) and related forms.

相应的积分结果常在 A-Level 题目中出现，尤其是涉及 1 / √(x² + 1) 及其相关形式的题目。

∫ sinh x dx = cosh x + C, ∫ cosh x dx = sinh x + C

∫ sech² x dx = tanh x + C

Using the logarithmic forms of inverse functions, the following standard integrals are obtained:

利用反函数的对数形式，可得到以下标准积分：

∫ 1/√(x² + 1) dx = arsinh x + C

∫ 1/√(x² − 1) dx = arcosh x + C, x > 1

∫ 1/(1 − x²) dx = artanh x + C, |x| < 1

8. Solving Equations with Hyperbolic Functions | 解双曲函数方程

There are two common methods for solving equations involving hyperbolic functions: using identities to reduce the equation to a single hyperbolic function, or rewriting all functions in terms of exp(x).

解含双曲函数的方程通常有两种方法：利用恒等式将方程化为单个双曲函数，或将所有函数改写为 exp(x) 的形式。

Example: Solve cosh 2x − 3 sinh x = 0 for real x.

示例：解方程 cosh 2x − 3 sinh x = 0，求实数 x。

Using cosh 2x = 1 + 2 sinh² x, the equation becomes 2 sinh² x + 1 − 3 sinh x = 0. Let u = sinh x, then 2u² − 3u + 1 = 0, giving u = 1/2 or u = 1.

利用 cosh 2x = 1 + 2 sinh² x，方程化为 2 sinh² x + 1 − 3 sinh x = 0。令 u = sinh x，得到 2u² − 3u + 1 = 0，故 u = 1/2 或 u = 1。

Thus sinh x = 1/2 gives x = arsinh(1/2) = ln(1/2 + √(5/4)); sinh x = 1 gives x = arsinh 1 = ln(1 + √2).

因此，sinh x = 1/2 时 x = arsinh(1/2) = ln(1/2 + √(5/4))；sinh x = 1 时 x = arsinh 1 = ln(1 + √2)。

When solving, always check that any solutions satisfy the original domain conditions, especially when using arcosh or artanh.

求解时，务必检查所有解是否满足原定义域条件，尤其在使用 arcosh 或 artanh 时。

9. Osborne’s Rule and Trigonometric Comparison | 奥斯本规则与三角对比

Osborne’s rule is a quick way to transform a trigonometric identity into a hyperbolic one. Replace each sin² term by −sinh², and change the sign of any product of two sine terms, then the identity holds.

奥斯本规则是将三角恒等式转换为双曲恒等式的快捷方法。将每个 sin² 项替换为 −sinh²，并改变任何两个正弦乘积项的符号，则恒等式仍然成立。

This rule works because the defining algebraic relationship between sin² and cos² differs from that between sinh² and cosh² by a sign.

该规则成立的原因在于 sin² 与 cos² 之间的代数关系，和 sinh² 与 cosh² 之间的代数关系仅相差一个符号。

Example: From cos 2x = cos² x − sin² x, applying the rule gives cosh 2x = cosh² x − (−sinh² x) = cosh² x + sinh² x.

示例：由 cos 2x = cos² x − sin² x，应用该规则得到 cosh 2x = cosh² x − (−sinh² x) = cosh² x + sinh² x。

This rule is not formally required by AQA but is a powerful memory aid. Always verify the final identity using exponential definitions if unsure.

该规则并非 AQA 官方要求，但是一个强大的记忆辅助工具。如果不确定，请始终用指数定义验证最终恒等式。

10. Summary and Exam Tips | 总结与考试建议

The key identities for AQA A-Level Further Mathematics are listed below. You should know them without hesitation and be able to derive them quickly.

以下是 AQA A-Level 进阶数学的关键恒等式。你应该熟练记忆，并能快速推导它们。

cosh² x − sinh² x = 1

cosh² x − sinh² x = 1
1 − tanh² x = sech² x

1 − tanh² x = sech² x
sinh 2x = 2 sinh x cosh x

sinh 2x = 2 sinh x cosh x
cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x

cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x
sinh(x ± y) = sinh x cosh y ± cosh x sinh y

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

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

When answering exam questions, write down the exponential definition at the start of any derivation. This earns method marks and helps prevent sign errors.

在考试答题时，请在推导开始时写出指数定义。这能获得方法分，并有助于避免符号错误。

Practice converting between the three forms of cosh 2x, as this is a common requirement in integration and equation-solving questions.

练习在 cosh 2x 的三种形式之间进行转换，这是积分和解方程题目中的常见要求。

Finally, remember that hyperbolic identities produce the same algebraic patterns as trigonometric identities but with different signs. Mastering these sign rules is the single most effective way to secure full marks.

最后，请记住双曲恒等式与三角恒等式具有相同的代数模式，但符号不同。掌握这些符号规则是获得满分最有效的方法。

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Hyperbolic Identities | 双曲恒等式