📚 Numerical Values of Hyperbolic Functions | 双曲函数的数值计算
Hyperbolic functions such as sinh x, cosh x and tanh x may look similar to trigonometric functions, but their numerical values come from exponential definitions rather than angles. In AQA A-Level Mathematics, students are expected to evaluate these functions, use exact forms involving natural logarithms, and apply the results in equations and identities.
双曲函数如 sinh x、cosh x 和 tanh x 看起来与三角函数相似，但它们的数值来自指数定义而不是角度。在 AQA A-Level 数学中，学生需要会计算这些函数的值，使用含有自然对数的精确形式，并在方程和恒等式中应用结果。
1. Definitions and Exponential Forms | 定义与指数形式
Hyperbolic functions are defined through the exponential function ex. For any real number x:
双曲函数通过指数函数 ex 定义。对任意实数 x：
sinh x = (ex − e−x) / 2
cosh x = (ex + e−x) / 2
The hyperbolic tangent is the ratio of sinh x to cosh x:
双曲正切是 sinh x 与 cosh x 的比值：
tanh x = sinh x / cosh x = (ex − e−x) / (ex + e−x)
Because these definitions contain only ex and e−x, any numerical value can be obtained without referring to angles.
因为这些定义只包含 ex 和 e−x，任何数值都可以不依赖角度直接求得。
2. Key Values at x = 0 | x = 0 处的关键值
Substituting x = 0 gives e0 = e−0 = 1, so the basic values are exact and simple.
代入 x = 0 得 e0 = e−0 = 1，因此基本值是精确且简单的。
sinh 0 = (1 − 1) / 2 = 0
cosh 0 = (1 + 1) / 2 = 1
tanh 0 = 0 / 1 = 0
The value cosh 0 = 1 is the minimum value of cosh x for all real x, because ex + e−x is always at least 2.
cosh 0 = 1 是 cosh x 对所有实数 x 的最小值，因为 ex + e−x 始终至少为 2。
3. Hand Calculation Using Exponential Forms | 使用指数形式手算求值
To find values without a hyperbolic key, use the exponential definitions and known approximations e ≈ 2.71828 and e−1 ≈ 0.36788.
如果计算器没有双曲键，可使用指数定义以及近似值 e ≈ 2.71828 和 e−1 ≈ 0.36788 来求值。
For x = 1:
当 x = 1 时：
sinh 1 = (e − e−1) / 2 ≈ (2.71828 − 0.36788) / 2 ≈ 1.17520
cosh 1 = (e + e−1) / 2 ≈ (2.71828 + 0.36788) / 2 ≈ 1.54308
tanh 1 = (e − e−1) / (e + e−1) ≈ 1.17520 / 1.54308 ≈ 0.76159
These three values appear frequently in AQA questions, so it is useful to recognise the approximate results sinh 1 ≈ 1.175, cosh 1 ≈ 1.543 and tanh 1 ≈ 0.762.
这三个值经常出现在 AQA 考题中，因此认识近似结果 sinh 1 ≈ 1.175、cosh 1 ≈ 1.543 和 tanh 1 ≈ 0.762 会很有帮助。
4. Calculator Evaluation and Accuracy | 计算器求值与精度
AQA questions often require calculator use, but you must know how to round and when an exact answer is expected. Hyperbolic calculations do not require switching between degree and radian modes because x is a real number, not an angle.
AQA 考题常要求使用计算器，但你必须知道如何舍入以及何时需要精确答案。双曲函数计算不需要在角度制和弧度制之间切换，因为 x 是实数，不是角度。
For example, evaluate cosh 2.1:
例如，计算 cosh 2.1：
cosh 2.1 = (e2.1 + e−2.1) / 2 ≈ (8.16617 + 0.12246) / 2 ≈ 4.14431
Rounded to three significant figures, cosh 2.1 ≈ 4.14.
保留三位有效数字，cosh 2.1 ≈ 4.14。
For negative inputs, use the symmetry rules sinh(−x) = −sinh x and cosh(−x) = cosh x. For instance, sinh(−0.4) = −sinh 0.4 ≈ −0.41075.
对于负输入，可使用对称性规则 sinh(−x) = −sinh x 和 cosh(−x) = cosh x。例如 sinh(−0.4) = −sinh 0.4 ≈ −0.41075。
5. Exact Values at ln 2 and ln 3 | 在 ln 2 和 ln 3 处的精确值
When x = ln a, the exponential terms simplify to a and 1/a, so exact rational values appear.
当 x = ln a 时，指数项化简为 a 和 1/a，因此会出现精确有理数值。
eln a = a and e−ln a = 1 / a
| x | sinh x | cosh x | tanh x |
|---|---|---|---|
| ln 2 | 3/4 | 5/4 | 3/5 |
| ln 3 | 4/3 | 5/3 | 4/5 |
For x = ln 2, for example:
例如当 x = ln 2 时：
sinh(ln 2) = (2 − 1/2) / 2 = 3/4
cosh(ln 2) = (2 + 1/2) / 2 = 5/4
tanh(ln 2) = (2 − 1/2) / (2 + 1/2) = 3/5
Exact answers like these are preferred in AQA when a question asks for the value in its simplest form.
当题目要求以最简形式给出值时，像这样的精确答案是 AQA 更偏好的。
6. Hyperbolic Identities as Numerical Checks | 双曲恒等式用于数值检验
The fundamental identity cosh²x − sinh²x = 1 is useful for checking calculator work.
基本恒等式 cosh²x − sinh²x = 1 可用于检查计算器结果。
At x = 1, cosh 1 ≈ 1.54308 and sinh
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