Numerical Values of Hyperbolic Functions | 双曲函数的数值计算

📚 Numerical Values of Hyperbolic Functions | 双曲函数的数值计算

Hyperbolic functions appear throughout mathematics and physics, and AQA A-Level questions frequently ask you to find exact numerical values of sinh, cosh and tanh at specific points. Because these functions are built from exponential functions, you can evaluate many exact values by hand using nothing more than the laws of indices.

双曲函数在数学与物理中广泛出现，AQA A-Level 试题经常要求你求出 sinh、cosh、tanh 在特定点处的精确数值。由于这些函数由指数函数构造而成，许多精确值仅需借助指数运算法则即可手算得出。

A clear grasp of the definitions, the identity cosh²x − sinh²x = 1, and the connection to logarithms will allow you to convert almost any hyperbolic equation into a manageable expression. This article develops each skill step by step, with exam-style examples throughout.

清晰掌握定义、恒等式 cosh²x − sinh²x = 1 以及与对数的联系，你就能把几乎所有的双曲方程转化为易于处理的形式。本文循序渐进地讲解各项技能，并配以考试风格例题。

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1. Exponential Definitions | 指数定义

For every real number x, the three main hyperbolic functions are defined directly in terms of exponential functions:

对任意实数 x，三个主要双曲函数均由指数函数直接定义：

sinh x = (eˣ − e⁻ˣ) / 2

cosh x = (eˣ + e⁻ˣ) / 2

tanh x = sinh x / cosh x = (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ)

Notice that e⁻ˣ means 1/eˣ, so every value of these functions is obtained by substituting powers of e. Because eˣ is always positive, cosh x is always greater than or equal to 1, and tanh x always lies between −1 and 1.

注意 e⁻ˣ 即 1/eˣ，因此这些函数的值都由 e 的幂代入得到。由于 eˣ 恒为正，所以 cosh x 始终大于或等于 1，而 tanh x 始终位于 −1 与 1 之间。

The single most useful substitution is x = ln k, because then eˣ = k and e⁻ˣ = 1/k exactly. This is why exact values of hyperbolic functions at logarithms of integers are so accessible.

最有用的代入是 x = ln k，因为此时 eˣ = k 且 e⁻ˣ = 1/k 为精确值。这正是整数对数点处的双曲函数值易于精确计算的原因。

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2. Exact Values at Key Points | 关键点处的精确值

We begin with the simplest point x = 0. Since e⁰ = 1, we obtain:

先从最简单的点 x = 0 开始。由于 e⁰ = 1，我们得到：

sinh 0 = (1 − 1) / 2 = 0, cosh 0 = (1 + 1) / 2 = 1, tanh 0 = 0

Now consider x = ln 2. We use e^(ln 2) = 2 and e^(−ln 2) = 1/2:

现在考虑 x = ln 2。我们使用 e^(ln 2) = 2 以及 e^(−ln 2) = 1/2：

sinh(ln 2) = (2 − 1/2) / 2 = (3/2) / 2 = 3/4

cosh(ln 2) = (2 + 1/2) / 2 = (5/2) / 2 = 5/4

tanh(ln 2) = (3/4) / (5/4) = 3/5

Repeating the same substitution for x = ln 3 gives a neat pattern:

对 x = ln 3 重复同样的代入，得到一组整齐的数值：

sinh(ln 3) = (3 − 1/3) / 2 = (8/3) / 2 = 4/3

cosh(ln 3) = (3 + 1/3) / 2 = (10/3) / 2 = 5/3

tanh(ln 3) = (4/3) / (5/3) = 4/5

These results are worth memorising, as they appear frequently in AQA papers. The table below summarises the most common exact values:

这些结果值得牢记，它们在 AQA 试卷中经常出现。下表汇总了最常见的精确值：

x	sinh x	cosh x	tanh x
0	0	1	0
ln 2	3/4	5/4	3/5
ln 3	4/3	5/3	4/5

Notice that 3/4, 5/4 and 4/3, 5/3 are examples of Pythagorean fractions: the pairs satisfy (3/4)² + 1 = (5/4)² and (4/3)² + 1 = (5/3)². This connection will reappear when we use the fundamental identity.

注意 3/4、5/4 以及 4/3、5/3 都是毕达哥拉斯分数：这两对满足 (3/4)² + 1 = (5/4)² 和 (4/3)² + 1 = (5/3)²。这种联系将在使用基本恒等式时再次出现。

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3. Parity and Negative Arguments | 奇偶性与负自变量

Because of the forms of the definitions, the hyperbolic functions have simple symmetry properties:

由于定义式的形式特点，双曲函数具有简单的对称性质：

sinh(−x) = −sinh x, cosh(−x) = cosh x, tanh(−x) = −tanh x

Thus sinh and tanh are odd functions while cosh is an even function. This means that a negative logarithm argument can be handled immediately:

因此 sinh 与 tanh 是奇函数，而 cosh 是偶函数。这意味着负对数自变量可以立即处理：

sinh(−ln 2) = −3/4, cosh(−ln 2) = 5/4, tanh(−ln 3) = −4/5

Graphically, y = cosh x is symmetric about the y-axis, while y = sinh x and y = tanh x have rotational symmetry about the origin. This symmetry is useful in solving equations: if x is a solution of an even equation, then −x is also a solution.

从图像上看，y = cosh x 关于 y 轴对称，而 y = sinh x 与 y = tanh x 关于原点中心对称。这种对称性在解方程时很有用：若 x 是某个偶

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