📚 Hyperbolic Functions | 双曲函数 考点精讲
Hyperbolic functions are a set of exponential-based functions that arise naturally in many areas of mathematics and physics. They share remarkable similarities with trigonometric functions but are defined using the hyperbola rather than the circle. This article covers the key concepts and techniques required for IB and OCR examinations.
双曲函数是一组基于指数函数的函数，在数学和物理的许多领域中自然出现。它们与三角函数有显著的相似之处，但是基于双曲线而非圆来定义。本文涵盖 IB 和 OCR 考试所需的关键概念与技巧。
1. Definitions and Basic Properties | 定义与基本性质
The hyperbolic sine and cosine are defined from the exponential function: sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2.
双曲正弦和双曲余弦由指数函数定义：sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2。
The hyperbolic tangent is tanh x = sinh x / cosh x = (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ).
双曲正切为 tanh x = sinh x / cosh x = (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ)。
The reciprocal functions are coth x = 1/tanh x, sech x = 1/cosh x, and csch x = 1/sinh x.
倒数函数为 coth x = 1/tanh x, sech x = 1/cosh x, csch x =
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