📚 GCSE Maths: Hyperbolic Functions – Key Exam Points | GCSE 数学：双曲函数 考点精讲
Hyperbolic functions are introduced in the IGCSE Additional Mathematics syllabus and are studied in far greater depth in A Level Further Mathematics. They are built directly from the exponential function eˣ, so every result you meet can be proved from first principles rather than memorised. They behave like trigonometric functions in many ways, but three differences matter in exams: a minus sign in the fundamental identity, no periodicity at all, and a range for cosh x that starts at 1 instead of -1.
双曲函数出现在剑桥 IGCSE 附加数学大纲中，并在 A Level 进阶数学里深入学习。它们直接由指数函数 eˣ 构造，因此你遇到的每一个结论都可以从最基础的定义推导出来，而不必死记硬背。它们在许多方面与三角函数相似，但有三个差异在考试中至关重要：主恒等式中的负号、完全没有周期性、以及 cosh x 的值域从 1 开始而不是 -1。
1. Definition of Hyperbolic Functions | 双曲函数的定义
The hyperbolic sine and hyperbolic cosine are defined as combinations of eˣ and e⁻ˣ. Everything else in this topic follows from these two definitions, so learn them exactly as written.
双曲正弦与双曲余弦定义为 eˣ 与 e⁻ˣ 的组合。本主题中其余所有内容都由这两个定义推出，因此必须一字不差地记住它们。
sinh x = (eˣ − e⁻ˣ) / 2 cosh x = (eˣ + e⁻ˣ) / 2
The hyperbolic tangent is the ratio of the two, and the three reciprocal functions are defined in the same way as sec, cosec and cot.
双曲正切是两者的比值，三个倒数函数则与 sec、cosec、cot 的定义方式完全一致。
tanh x = sinh x / cosh x = (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ)
x sech x = 1 / cosh x cosech x = 1 / sinh x
ok like the Euler formulas for sine and cosine but without the imaginary unit i. Because eˣ By contrast, sinh x equals zero only at x = 0, so coth x and cosech x are undefined there.
这些定义看起来像正弦和余弦的欧拉公式，但没有虚数单位 i。由于 eˣ 恒为正，cosh x 永远不会小于 1，所以 tanh x 和 sech x 对任意实数 x 都有定义。相反，sinh x 只在 x = 0 处等于零，因此 coth x 和 cosech x 在该点无定义。
2. Basic Identities | 基本恒等式
The single most important identity in the whole topic is the hyperbolic version of the Pythagorean identity. Notice that the plus sign becomes a minus sign.
整个主题中最重要的恒等式是毕达哥拉斯恒等式的双曲版本。注意加号变成了减号。
cosh²x − sinh²x = 1
You can prove it in one line by substituting the exponential definitions and using the difference of two squares. The proof is a standard exam question, so practise writing it out cleanly.
只需代入指数定义并使用平方差公式，一行即可证明。这个证明是常见的考题，务必练习到能干净利落地写出全过程。
[(eˣ + e⁻ˣ)² − (eˣ − e⁻ˣ)²] / 4 = [4 eˣ e⁻ˣ] / 4 = 1
Dividing the fundamental identity by cosh²x or by sinh²x produces two useful variants that turn up in integration questions.
把主恒等式分别除以 cosh²x 和 sinh²x，可得到两个在积分题中很有用的变形。
sech²x = 1 − tanh²x cosech²x = coth²x − 1
+ tan²x. The only change is the sign, and that sign change is exactly what examiners test.
把第一个式子与三角结果 sec²x = 1 + tan²x 对比。唯一的区别就是符号，而恰恰是这个符号变化是考官最爱考的地方。
3. Double-Angle and Half-Angle Formulas | 二倍角与半角公式
The double-argument formulas are used constantly when reducing a hyperbolic equation to a quadratic. Learn all three versions of cosh 2x, because the exam may require whichever one makes the algebra shortest.
在把双曲方程化为二次方程时，二倍角公式会被反复使用。cosh 2x 的三种形式都要掌握，因为考题可能需要你选用让运算最短的那一种。
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh²x + sinh²x = 2cosh²x − 1 = 1 + 2sinh²x
tanh 2x = 2 tanh x / (1 + tanh²x)
Be careful with tanh 2x: the denominator carries a plus sign, whereas the trigonometric tan 2x has a minus sign in the denominator. This is a classic trap in multiple-choice and short-answer questions.
tanh 2x 要特别小心：分母是加号，而三角函数的 tan 2x 分母是减号。这是选择题和简答题中的经典陷阱。
Rearranging the cosh 2x forms gives the half-argument results, which are the hyperbolic equivalents of the power-reduction formulas.
把 cosh 2x 的几种形式重新整理，便得到半角结果，它们是三角降幂公式的双曲对应物。
²(x/2) = (cosh x − 1) / 2 cosh²(x/2) = (cosh x + 1) / 2
Triple-argument results occasionally appear in stretch questions and are worth recognising rather than deriving under time pressure.
三倍角结果偶尔出现在拔高题中，值得直接记住，而不是在考试时间压力下现推。
h 3x = 3 sinh x + 4 sinh³x cosh 3x = 4cosh³x − 3 cosh x
4. Addition Formulas and Osborn's Rule | 加法公式与奥斯本法则
The addition formulas mirror the trigonometric ones, with one systematic sign change in every product of two sines.
加法公式与三角函数的对应公式几乎一样，只是在每一项含两个正弦之积的地方会系统性地变号。
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
tanh(x + y) = (tanh x + tanh y) / (1 + tanh x tanh y)
Osborn's rule lets you convert almost any trigonometric identity into a hyperbolic one without re-deriving it. Replace every sin with sinh, every cos with cosh, and change the sign of any term that contains a product of two sines.
奥斯本法则让你无需重新推导，就能把几乎任何三角恒等式转成双曲恒等式。做法是：把每个 sin 换成 sinh，每个 cos 换成 cosh，并对任何含两个正弦之积的项改变符号。
| cos 2x = 1 − 2sin²x | cosh 2x = 1 + 2sinh²x |
| cos(A + B) = cos A cos B − sin A sin B | cosh(x + y) = cosh x cosh y + sinh x sinh y |
| sin(A + B) = sin A cos B + cos A sin B | sinh(x + y) = sinh x cosh y + cosh x sinh y |
5. Graphs, Symmetry, Domain and Range | 图像、奇偶性、定义域与值域
Sketching the three basic graphs is a quick mark and a good way to check your algebra. sinh x is an odd function passing through the origin and increasing strictly, behaving like eˣ/2 for large positive x and like −e⁻ˣ/2 for large negative x.
画出三个基本图像既能快速拿分，也能用来检验你的代数结果。sinh x 是奇函数，过原点且严格递增，x 很大时近似 eˣ/2，x 很负时近似 −e⁻ˣ/2。
es through the origin, and is squeezed between the horizontal asymptotes y = 1 and y = −1.
cosh x 是偶函数，在 x = 0 处取最小值 1，其图像就是悬挂链条的形状。tanh x 是奇函数，过原点，并被两条水平渐近线 y = 1 与 y = −1 夹住。
| Function | Symmetry | Domain | Range |
| sinh x | odd | all real x | all real y |
| cosh x | even | all real x | y ≥ 1 |
| tanh x | style="border:1px solid #
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