GCSE Maths: Hyperbolic Functions – Key Exam Points | GCSE 数学：双曲函数 考点精讲

📚 GCSE Maths: Hyperbolic Functions – Key Exam Points | GCSE 数学：双曲函数 考点精讲

Hyperbolic functions appear in the IGCSE Additional Mathematics syllabus. They are defined using exponential functions and share many properties with trigonometric functions but with important differences. Mastering sinh x, cosh x and tanh x, along with their identities and equations, is essential for top marks.

双曲函数是剑桥 IGCSE 附加数学的重要内容。它们由指数函数定义，与三角函数有许多类似性质，也存在关键差异。掌握 sinh x、cosh x 和 tanh x 及其恒等式与方程，是拿高分的关键。

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

The hyperbolic sine and cosine are defined by: sinh x = (eˣ – e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2. The hyperbolic tangent is then tanh x = sinh x / cosh x = (eˣ – e⁻ˣ)/(eˣ + e⁻ˣ).

双曲正弦和双曲余弦的定义为：sinh x = (eˣ – e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2。进而双曲正切为 tanh x = sinh x / cosh x = (eˣ – e⁻ˣ)/(eˣ + e⁻ˣ)。

These definitions are similar to the Euler formulas for sine and cosine but without the imaginary unit i. Unlike sine and cosine, sinh and cosh are not periodic.

这些定义类似于正弦和余弦的欧拉公式，但没有虚数单位 i。与正弦和余弦不同，双曲正弦和双曲余弦不是周期函数。

2. Basic Identities | 基本恒等式

The fundamental identity is cosh²x – sinh²x = 1. This is analogous to cos²x + sin²x = 1 but with a minus sign. It can be proved by substituting the exponential definitions.

最基本的恒等式是 cosh²x – sinh²x = 1。它类似于 cos²x + sin²x = 1，但有一个负号。可代入指数定义进行证明。

Other essential identities include the double-argument formulas: sinh 2x = 2

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