📚 Hyperbolic Identities | 双曲恒等式
Hyperbolic identities are a central part of AQA Further Mathematics. They look similar to trigonometric identities, but the signs are often different. Understanding how to derive and apply them will help you solve equations, differentiate and integrate hyperbolic functions with confidence.
双曲恒等式是 AQA 进阶数学的核心内容。它们看起来与三角恒等式相似，但符号常常不同。理解如何推导和应用这些恒等式，将帮助你自信地解方程、求导和积分双曲函数。
1. What Are Hyperbolic Functions? | 什么是双曲函数？
Hyperbolic functions are built from the exponential function. The two fundamental functions are sinh x and cosh x, defined by:
双曲函数由指数函数构造而成。两个基本函数是 sinh x 和 cosh x，其定义为：
sinh x = (eˣ - e⁻ˣ)/2
cosh x = (eˣ + e⁻ˣ)/2
The third common function, tanh x, is the quotient of sinh x and cosh x. It can also be written directly in terms of eˣ.
第三个常用函数 tanh x 是 sinh x 与 cosh x 的商。它也可以直接用 eˣ 表示。
tanh x = sinh x / cosh x = (eˣ - e⁻ˣ)/(eˣ + e⁻ˣ)
ange [1, ∞), and tanh x has range (-1, 1). These ranges are useful when solving equations.
根据这些定义，sinh x 的定义域和值域均为全体实数，cosh x 的值域为 [1, ∞)，tanh x 的值域为 (-1, 1)。这些值域在解方程时非常有用。
2. Osborn's Rule | 奥斯本法则
Osborn's rule is a quick way to convert a trigonometric identity into the corresponding hyperbolic identity. Start with a standard trigonometric identity, replace sin with sinh and cos with cosh, and change the sign of any term containing a product of two sinh functions.
奥斯本法则是将三角恒等式转换为相应双曲恒等式的快捷方法。从一个标准三角恒等式出发，将 sin 替换为 sinh，将 cos 替换为 cosh，并将任何含有两个 sinh 乘积的项改变符号。
- Start with cos² x + sin² x = 1.
从 cos² x + sin² x = 1 开始。 - Replace cos by cosh and sin by sinh, then change the sign of the sin² term.
将 cos 替换为 cosh，将 sin 替换为 sinh，然后改变 sin² 项的符号。 - This gives cosh² x - sinh² x = 1.
于是得到 cosh² x - sinh² x = 1。
Do not treat Osborn' se sin(ix) = i sinh x and cos(ix) = cosh x, so a product of two sines contributes i² = -1.
不要将奥斯本法则当作猜测。它成立的原因是 sin(ix) = i sinh x 且 cos(ix) = cosh x，因此两个正弦的乘积会产生 i² = -1。
| Trigonometric identity | Hyperbolic identity |
|---|---|
| cos² x + sin² x = 1 | cosh² x - sinh² x = 1 |
| 1 + tan² x = sec² x | 1 - tanh² x = sech² x |
| 1 + cot² x = cosec² x | coth² x - 1 = csch² x |
3. The Fundamental Identity | 基本恒等式
milar to the trigonometric identity cos² x + sin² x = 1, but the negative sign is crucial.
最重要的双曲恒等式是 cosh² x - sinh² x = 1。它类似于三角恒等式 cos² x + sin² x = 1，但其中的负号至关重要。
To prove it, substitute the exponential definitions and expand the squares carefully.
要证明它，可以代入指数定义并仔细展开平方。
cosh² x - sinh² x = [(eˣ + e⁻ˣ)/2]² - [(eˣ - e⁻ˣ)/2]²
The numerator simplifies to 4, so the value is 1. This identity is used constantly in simplifying expressions and solving equations.
分子化简为 4，因此结果为 1。这个恒等式在化简表达式和解方程时经常使用。
cosh x is always at least 1: from cosh² x = 1 + sinh² x ≥ 1, we get cosh x ≥ 1 for real x.
它还说明了为什么 cosh x 总是大于或等于 1：由 cosh² x = 1 + sinh² x ≥ 1，可得对于实数 x 有 cosh x ≥ 1。
4. Reciprocal and Quotient Identities | 倒数与商恒等式
The reciprocal hyperbolic functions are sech x, csch x and coth x. They are defined as follows:
倒数双曲函数分别是 sech x、csch x 和 coth x。它们的定义如下：
sech x = 1/cosh x, csch x = 1/sinh x, coth x = 1/tanh x
The quotient identity for tanh x has already been given, but it is worth restating because it is used in differentiation and equation solving:
tanh x 的商恒等式前面已经给出，但值得再次强调，因为它在求导和解方程中都会用到：
tanh x = sinh x / cosh x, coth x = cosh x / sinh x
These definitions allow you to rewrite any hyperbolic expression in terms of sinh x and cosh x. This is often the first step when proving more complicated identities.
这些定义使你可以将任何双曲表达式用 sinh x 和 cosh x 重写。这通常是证明更复杂恒等式的第一步。
5. Symmetry and Parity | 对称性与奇偶性
Hyperbolic functions have simple symmetry properties. The function sinh x is odd, cosh x is even, and tanh x is odd.
双曲函数具有简单的对称性。sinh x 是奇函数，cosh x 是偶函数，tanh x 是奇函数。
sinh(-x) = -sinh x, cosh(-x) = cosh x, tanh(-x) = -tanh x
These properties follow directly from the exponential definitions. For example, replacing x by -x in sinh x gives (e⁻ˣ - eˣ)/2, which is the negative of sinh x.
这些性质直接来自指数定义。例如，在 sinh x 中将 x 替换为 -x 得到 (e⁻ˣ - eˣ)/2，它正是 sinh x 的相反数。
simplify expressions such as sinh(-a) + cosh(-a) = -sinh a + cosh a, which is exactly e⁻ᵃ.
了解每个函数的奇偶性有助于化简表达式。例如 sinh(-a) + cosh(-a) = -sinh a + cosh a，而它恰好等于 e⁻ᵃ。
6. Pythagorean-Type Identities | 毕达哥拉斯型恒等式
= 1, you can divide by cosh² x or sinh² x to obtain two more Pythagorean-type identities.
从 cosh² x - sinh² x = 1 出发，你可以除以 cosh² x 或 sinh² x，得到另外两个毕达哥拉斯型恒等式。
Dividing by cosh² x gives the identity involving tanh x and sech x:
除以 cosh² x 得到含有 tanh x 和 sech x 的恒等式：
1 - tanh² x = sech² x
Dividing by sinh² x gives the identity involving coth x and csch x:
除以 sinh² x 得到含有 coth x 和 csch x 的恒等式：
coth² x - 1 = csch² x
These forms are especially helpful when an equation contains tanh x and sech x, or coth x and csch x. They allow you to reduce the number of different functions in the problem.
当方程中含有 tanh x 和 sech x，或 coth x 和 csch x 时，这些形式尤其有用。它们可以帮助你减少问题中不同函数的数量。
7. Addition Formulae | 加法公式
hyperbolic functions resemble the trigonometric ones, but the signs differ in cosh(A ± B).
双曲函数的加法公式与三角加法公式相似，但 cosh(A ± B) 中的符号不同。
sinh(A + B) = sinh A cosh B + cosh A sinh B
sinh(A - B) = sinh A cosh B - cosh A sinh B
For cosh, the sign in front of the product of sinh terms is positive, not negative:
对于 cosh，两个 sinh 乘积前面的符号是正号，而不是负号：
cosh(A + B) = cosh A cosh B + sinh A sinh B
cosh(A - B) = cosh A cosh B - sinh A sinh B
The tanh addition formula follows by dividing sinh(A + B) by cosh(A + B).
tanh 的加法公式可以通过将 sinh(A + B) 除以 cosh(A + B) 得到。
tanh(A + B) = (tanh A + tanh B)/(1 + tanh A tanh B)
These formulae are used to derive double-angle identities and to prove other hyperbolic relations.
这些公式用于推导双角恒等式以及证明其他双曲关系。
8. Double-Angle Identities | 双角恒等式
formulae gives the double-angle identities. They are particularly useful for integration.
在加法公式中令 A = B = x，即可得到双角恒等式。它们在积分中特别有用。
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh² x + sinh² x
x = cosh² x - 1, the cosh double-angle identity can be written in three equivalent forms:
利用 cosh² x = 1 + sinh² x 和 sinh² x = cosh² x - 1，cosh 的双角恒等式可以写成三种等价形式：
cosh 2x = 2 cosh² x - 1 = 1 + 2 sinh² x
The tanh double-angle identity is:
tanh 的双角恒等式为：
tanh 2x = 2 tanh x/(1 + tanh² x)
These rearrangements are essential when you need to replace squares of hyperbolic functions with a single hyperbolic function of double angle.
当你需要将双曲函数的平方替换为单一的双角双曲函数时，这些变形至关重要。
9. Half-Angle and Product Identities | 半角与积化和差恒等式
The half-angle identities come from rearranging the double-angle formula for cosh 2x. They are very common in integration.
半角恒等式来自 cosh 2x 双角公式的变形。它们在积分中非常常见。
sinh²(x/2) = (cosh x - 1)/2
cosh²(x/2) = (cosh x + 1)/2
Product-to-sum identities convert products of hyperbolic functions into sums. They are analogous to the trigonometric product-to-sum formulae.
积化和差恒等式将双曲函数的乘积转换为和差。它们类似于三角函数的积化和差公式。
sinh A cosh B = ½[sinh(A + B) + sinh(A - B)]
cosh A cosh B = ½[cosh(A + B) + cosh(A - B)]
sinh A sinh B = ½[cosh(A + B) - cosh(A - B)]
These identities are particularly useful when integrating products such as sinh 3x cosh x. You can convert the product into a sum and integrate each term separately.
这些恒等式在积分 sinh 3x cosh x 等乘积时特别有用。你可以将乘积转换为和，然后逐项积分。
10. Solving Hyperbolic Equations | 解双曲方程
Hyperbolic equations can usually be solved by one of two methods: rewriting everything in exponential form, or using an identity to reduce the equation to a single hyperbolic function.
双曲方程通常可以通过两种方法之一求解：将所有项改写为指数形式，或使用恒等式将方程化为只含一个双曲函数的形式。
Example 1: Solve 3 sinh x - cosh x = 1.
例 1：解方程 3 sinh x - cosh x = 1。
Using exponential definitions gives 3(eˣ - e⁻ˣ)/2 - (eˣ + e⁻ˣ)/2 = 1. Simplifying gives eˣ - 2e⁻ˣ = 1, then multiplying by eˣ gives e²ˣ - eˣ - 2 = 0.
使用指数定义可得 3(eˣ - e⁻ˣ)/2 - (eˣ + e⁻ˣ)/2 = 1。化简得 eˣ - 2e⁻ˣ = 1，再乘以 eˣ 得 e²ˣ - eˣ - 2 = 0。
This factors as (eˣ - 2)(eˣ + 1) = 0. Since eˣ is always positive, eˣ = 2, so x = ln 2.
该方程可分解为 (eˣ - 2)(eˣ + 1) = 0。由于 eˣ 恒为正，因此 e
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