📚 Logarithmic Form of Inverse Hyperbolic Functions | 反双曲函数的对数形式
In A-Level Mathematics, the hyperbolic functions sinh x, cosh x and tanh x are defined in terms of exponential functions. Their inverse functions, written arsinh x, arcosh x and artanh x, reverse these hyperbolic operations. This article explains how each inverse hyperbolic function can be written exactly as a natural logarithm, a skill that is regularly examined by AQA.
在 A-Level 数学中，双曲函数 sinh x、cosh x 和 tanh x 都是用指数函数定义的。它们的反函数写作 arsinh x、arcosh x 和 artanh x，可以逆转这些双曲运算。本文将讲解如何把每个反双曲函数精确写成自然对数形式，这是 AQA 考试中经常考查的技能。
1. Inverse Hyperbolic Functions: What They Are | 什么是反双曲函数
The notation arsinh x means 'area hyperbolic sine of x'. It is also written as sinh⁻¹ x in some books, but AQA prefers arsinh x to avoid confusion with the reciprocal of sinh x. The same applies to arcosh x and artanh x.
arsinh x 表示 '反双曲正弦'。有些教材也写作 sinh⁻¹ x，但 AQA 更倾向于使用 arsinh x，以避免与 sinh x 的倒数混淆。arcosh x 和 artanh x 同理。
These functions are not the same as the ordinary inverse trigonometric functions arcsin x, arccos x and arctan x. The inverse hyperbolic functions can be expressed using natural logarithms because their original definitions involve exponentials.
这些函数与普通的反三角函数 arcsin x、arccos x 和 arctan x 不同。由于反双曲函数的原始定义涉及指数函数，因此它们可以用自然对数来表示。
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