📚 Definitions of Hyperbolic Functions | 双曲函数的定义
Hyperbolic functions are combinations of exponential functions that appear frequently in AQA Further Mathematics. They share many algebraic similarities with trigonometric functions, but their definitions are based on eˣ and e⁻ˣ rather than on a circle.
双曲函数是由指数函数组合而成的一类函数，在 AQA 进阶数学中经常出现。它们与三角函数在代数性质上有许多相似之处，但其定义基于 eˣ 和 e⁻ˣ，而不是基于圆。
1. Defining sinh x and cosh x | 定义 sinh x 与 cosh x
The two fundamental hyperbolic functions are the hyperbolic sine and hyperbolic cosine. For all real x:
两个最基本的双曲函数是双曲正弦和双曲余弦。对于所有实数 x：
sinh x = (eˣ − e⁻ˣ) / 2
cosh x = (eˣ + e⁻ˣ) / 2
These definitions show that sinh x is an odd function and cosh x is an even function.
这些定义表明 sinh x 是奇函数，而 cosh x 是偶函数。
2. Defining tanh x | 定义 tanh x
The hyperbolic tangent is defined as the quotient of sinh x and cosh x. Because cosh x is never zero, tanh x is defined for all real x.
双曲正切定义为 sinh x 与 cosh x 的商。由于 cosh x 始终不为零，因此 tanh x 对所有实数 x 都有定义。
tanh x = sinh x / cosh x = (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ) = (e²ˣ − 1) / (e²ˣ + 1)
Multiplying numerator and denominator by eˣ gives the final rational form in e²ˣ.
将分子和分母同时乘以 eˣ，就得到用 e²ˣ 表示的有理分式形式。
3. Reciprocal hyperbolic functions | 倒数双曲函数
The reciprocal functions are sech x, cosech x and coth x. They are defined in the same way as reciprocal trigonometric functions, but domain restrictions apply where the denominator is zero.
倒数双曲函数包括 sech x、cosech x 和 coth x。它们的定义方式与倒数三角函数类似，但在分母为零处需要限制定义域。
- sech x = 1 / cosh x = 2 / (eˣ + e⁻ˣ) — 双曲正割
- cosech x = 1 / sinh x = 2 / (eˣ − e⁻ˣ) — 双曲余割，x ≠ 0
- coth x = 1 / tanh x = cosh x / sinh x = (eˣ + e⁻ˣ) / (eˣ − e⁻ˣ) — 双曲余切，x ≠ 0
Note that coth x can also be written as (e²ˣ + 1) / (e²ˣ − 1) after multiplying by eˣ.
注意，coth x 也可以在乘以 eˣ 后写成 (e²ˣ + 1) / (e²ˣ − 1)。
4. Domains and ranges | 定义域与值域
Knowing the domain and range of each hyperbolic function is essential for sketching graphs and solving equations.
了解每个双曲函数的定义域和值域，对画图和求解方程都至关重要。
| Function 函数 | Domain 定义域 | Range 值域 |
|---|---|---|
| sinh x | ℝ | ℝ |
| cosh x | ℝ | [1, ∞) |
| tanh x | ℝ | (−1, 1) |
| sech x | ℝ | (0, 1] |
| cosech x | ℝ \ {0} | ℝ \ {0} |
| coth x | ℝ \ {0} | (−∞, −1) ∪ (1, ∞) |
important values to remember are that cosh x ≥ 1 and −1 < tanh x < 1 for all real x.
最需要记住的是：对所有实数 x，cosh x ≥ 1，且 −1 < tanh x < 1。
5. Graphs and key features | 图像与关键特征
the origin and is increasing for all x. It has no asymptotes and covers all real y-values.
y = sinh x 的图像经过原点，并且在所有实数范围内单调递增。它没有渐近线，值域覆盖所有实数。
The graph of y = cos
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