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

In A-Level Further Mathematics, hyperbolic functions are defined using the exponential function. They are not simply “trigonometric functions in disguise”; their numerical values arise naturally from combinations of \(e^x\) and \(e^{-x}\), and they play a key role in calculus, differential equations and integration.

在 A-Level 进阶数学中，双曲函数是用指数函数来定义的。它们并不是仅仅“伪装成三角函数”的函数；其数值来源于 eˣ 与 e⁻ˣ 的组合，并且在微积分、微分方程和积分中扮演着重要角色。

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

The two basic hyperbolic functions are defined as follows:

两个基本双曲函数的定义如下：

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

Consequently, the other four hyperbolic functions are:

因此，其余四个双曲函数为：

tanh x = sinh x / cosh x = (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ), coth x = cosh x / sinh x, sech x = 1 / cosh x, cosech x = 1 / sinh x

These definitions are the starting point for every numerical evaluation. To compute a value, we first express the function in terms of exponentials, then substitute the given x.

这些定义是所有数值计算的起点。要计算一个值，我们首先将函数写成指数形式，然后代入给定的 x。

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

Knowing exact values at simple arguments helps in sketching graphs and checking calculator results.

掌握几个简单自变量的精确值，有助于画图以及检验计算器结果。

sinh 0 = (1 − 1)/2 = 0

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

cosh 0 = (1 + 1)/2 = 1
tanh 0 = 0

tanh 0 = 0
sinh(ln 2) = (2 − 1/2)/2 = 3/4

sinh(ln 2) = (2 − 1/2)/2 = 3/4
cosh(ln 2) = (2 + 1/2)/2 = 5/4

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

For example, using e^(ln 2) = 2, we obtain sinh(ln 2) = 3/4 exactly. This shows that hyperbolic functions can give simple rational outputs for specific inputs.

例如，利用 e^(ln 2) = 2，可以得到 sinh(ln 2) = 3/4 精确值。这表明双曲函数在某些特定输入下会给出简单的有理数输出。

3. Approximate Values for Common Inputs | 常见输入的近似值

It is useful to memorise the following approximate values:

记住下列常见近似值很有帮助：

x	sinh x	cosh x	tanh x
0	0	1	0
0.5	0.5211	1.1276	0.4621
1	1.1752	1.5431	0.7616
2	3.6269	3.7622	0.9640

For negative arguments, remember that sinh(−x) = −sinh x and tanh(−x) = −tanh x, hence sinh(−1) ≈ −1.1752 and tanh(−1) ≈ −0.7616. By contrast, cosh(−x) = cosh x, so cosh(−1) ≈ 1.5431.

对于负自变量，要记住 sinh(−x) = −sinh x，tanh(−x) = −tanh x，因此 sinh(−1) ≈ −1.1752，tanh(−1) ≈ −0.7616。相反，cosh(−x) = cosh x，所以 cosh(−1) ≈ 1.5431。

4. Symmetry and Sign | 对称性与符号

Hyperbolic functions have distinct parity properties:

双曲函数具有不同的奇偶性质：

sinh x and tanh x are odd functions.

sinh x 与 tanh x 是奇函数。
cosh x is an even function.

cosh x 是偶函数。
sinh x is positive for x > 0 and negative for x < 0.

当 x > 0 时 sinh x 为正，当 x < 0 时 sinh x 为负。
cosh x is always at least 1, with minimum value 1 at x = 0.

cosh x 始终不小于 1，在 x = 0 处取得最小值 1。
tanh x lies between −1 and 1, approaching 1 as x → ∞.

tanh x 位于 −1 和 1 之间，当 x → ∞ 时趋近于 1。

These properties are vital when evaluating numerical expressions, because they allow us to reduce arguments to positive values before using a calculator.

这些性质在数值计算中非常重要，因为它们允许我们在使用计算器之前将自变量转化为正值。

5. Fundamental Identities | 基本恒等式

The most important identity is:

最重要的恒等式是：

cosh² x − sinh² x = 1

This identity is the hyperbolic analogue of sin² x + cos² x = 1, but with a minus sign. It can be used to find one hyperbolic function from another when the numerical value is known.

这个恒等式是 sin² x + cos² x = 1 的双曲类比，但符号为负。已知一个双曲函数的数值时，可以利用它求出另一个双曲函数。

For example, if sinh x = 1/2, then cosh² x = 1 + (1/2)² = 5/4, so cosh x = √5 / 2 ≈ 1.1180 (taking the positive root).

例如，若 sinh x = 1/2，则 cosh² x = 1 + (1/2)² = 5/4，因此 cosh x = √5 / 2 ≈ 1.1180（取正根）。

6. Related Identities for Other Functions | 其它函数的相关恒等式

From the definitions we can derive:

由定义可以推出：

1 − tanh² x = sech² x, coth² x − 1 = cosech² x

Also, the reciprocal relationships hold:

同时还有倒数关系：

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

These identities allow us to convert between numerical values. For instance, if tanh x = 0.8, then sech² x = 1 − 0.64 = 0.36, hence sech x = 0.6, so cosh x = 1/0.6 = 5/3.

这些恒等式允许我们在不同函数数值之间转换。例如，若 tanh x = 0.8，则 sech² x = 1 − 0.64 = 0.36，因此 sech x = 0.6，所以 cosh x = 1/0.6 = 5/3。

7. Using Exponential Form for Calculation | 利用指数形式进行计算

When a calculator is not available, hyperbolic functions can be evaluated directly from exponentials.

当没有计算器时，双曲函数可以直接从指数形式求得。

For x = 1, using e ≈ 2.7183:

对于 x = 1，利用 e ≈ 2.7183：

sinh 1 = (2.7183 − 0.3679) / 2 ≈ 1.1752, cosh 1 = (2.7183 + 0.3679) / 2 ≈ 1.5431

For large x, e⁻ˣ becomes extremely small, so sinh x ≈ cosh x ≈ eˣ/2. For example, sinh 3 ≈ (20.0855 − 0.0498)/2 ≈ 10.0179, while cosh 3 ≈ (20.0855 + 0.0498)/2 ≈ 10.0677.

当 x 较大时，e⁻ˣ 变得非常小，因此 sinh x ≈ cosh x ≈ eˣ/2。例如，sinh 3 ≈ (20.0855 − 0.0498)/2 ≈ 10.0179，而 cosh 3 ≈ (20.0855 + 0.0498)/2 ≈ 10.0677。

8. Solving Equations Involving Hyperbolic Functions | 求解含双曲函数的方程

To find x given a numerical value of a hyperbolic function, we often convert to exponential form.

若已知双曲函数的数值要求 x，通常将函数转化为指数形式。

For example, solve sinh x = 2. Using the definition:

例如，解方程 sinh x = 2。由定义：

(eˣ − e⁻ˣ)/2 = 2 ⇒ eˣ − e⁻ˣ = 4 ⇒ e²ˣ − 4eˣ − 1 = 0

Let y = eˣ > 0. Then y² − 4y − 1 = 0, so y = 2 + √5. Therefore x = ln(2 + √5) ≈ ln(4.236) ≈ 1.4436.

令 y = eˣ > 0，则 y² − 4y − 1 = 0，因此 y = 2 + √5。所以 x = ln(2 + √5) ≈ ln(4.236) ≈ 1.4436。

This method produces exact answers in terms of logarithms, which are often preferred over decimal approximations in examinations.

这种方法可以得到含对数的精确答案，在考试中通常比小数近似值更受青睐。

9. Inverse Hyperbolic Functions and Numerical Values | 反双曲函数及其数值

The inverse hyperbolic functions are defined using logarithms:

反双曲函数用对数定义：

arsinh x = ln(x + √(x² + 1)), arcosh x = ln(x + √(x² − 1)) (x ≥ 1), artanh x = ½ ln((1 + x)/(1 − x)) (−1 < x < 1)

These formulas allow exact evaluation. For instance, arcosh 3 = ln(3 + √8) = ln(3 + 2√2) ≈ ln(5.828) ≈ 1.7627.

这些公式可以进行精确计算。例如，arcosh 3 = ln(3 + √8) = ln(3 + 2√2) ≈ ln(5.828) ≈ 1.7627。

Notice that arcosh x always produces a non-negative value, because the principal value is chosen. In applied contexts, both positive and negative roots of cosh x = a may be required.

注意 arcosh x 总是给出非负值，因为取主值。在实际应用中，cosh x = a 的正负根可能都需要考虑。

10. Common Pitfalls and Exam Tips | 常见错误与考试提示

Avoid mixing hyperbolic identities with trigonometric ones.

避免将双曲恒等式与三角恒等式混淆。

cosh² x − sinh² x = 1, not cosh² x + sinh² x = 1.

cosh² x − sinh² x = 1，而不是 cosh² x + sinh² x = 1。
sinh 0 = 0, but cosh 0 = 1. Do not assume cosh 0 = 0.

sinh 0 = 0，但 cosh 0 = 1。不要认为 cosh 0 = 0。
tanh x is not equal to sin x / cos x; it is sinh x / cosh x.

tanh x 不等于 sin x / cos x；它是 sinh x / cosh x。
When using a calculator, select the correct mode: hyperbolic functions are usually labelled sinh, cosh and tanh.

使用计算器时，应选择正确的模式：双曲函数通常标注为 sinh、cosh 和 tanh。
Always check whether an answer should be exact (logarithmic form) or approximate (decimal form).

始终留意答案应当写成精确形式（对数形式）还是近似形式（小数形式）。

By mastering the definitions, identities and symmetry properties, you can confidently evaluate hyperbolic functions numerically and solve related equations in the AQA A-Level Further Mathematics examination.

通过掌握定义、恒等式和对称性，你就能自信地计算双曲函数的数值，并在 AQA A-Level 进阶数学考试中解决相关问题。

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