Hyperbolic Functions for KS3 | KS3 数学：双曲函数考点精讲

📚 Hyperbolic Functions for KS3 | KS3 数学：双曲函数考点精讲

Welcome to the fascinating world of hyperbolic functions! While these are often taught at A‑level or beyond, we can still explore their core ideas in a KS3‑friendly way. Hyperbolic functions – sinh, cosh, and tanh – describe the shape of a hanging chain, the growth of certain populations, and even appear in special relativity. This article will introduce their definitions, graphs, and key properties, building strong foundations for future study.

欢迎来到双曲函数的奇妙世界！虽然双曲函数通常在 A‑level 或更高阶段讲授，但我们完全可以用 KS3 也能理解的方式探索它们的核心思想。双曲正弦（sinh）、双曲余弦（cosh）和双曲正切（tanh）可以描述悬链线的形状、某些种群的增长，甚至出现在狭义相对论中。本文将介绍它们的定义、图像和重要性质，为将来的学习打下坚实基础。

1. What Are Hyperbolic Functions? | 什么是双曲函数？

Hyperbolic functions are exponential relatives of the ordinary trigonometric functions. Instead of being built from the unit circle x² + y² = 1, they are derived from the unit hyperbola x² – y² = 1. The three main ones are sinh (hyperbolic sine), cosh (hyperbolic cosine), and tanh (hyperbolic tangent).

双曲函数是指数函数与普通三角函数的“表亲”。它们并非源于单位圆 x² + y² = 1，而是源于单位双曲线 x² – y² = 1。三个主要函数是 sinh（双曲正弦）、cosh（双曲余弦）和 tanh（双曲正切）。

2. Definition of sinh x | 双曲正弦的定义

The hyperbolic sine function is defined using exponential functions: sinh x = (eˣ – e⁻ˣ) / 2. This formula gives sinh x for any real number x. It is an odd function, meaning sinh(–x) = –sinh x.

双曲正弦函数用指数定义：sinh x = (eˣ – e⁻ˣ) / 2。此公式对所有实数 x 都成立。它是一个奇函数，即 sinh(–x) = –sinh x。

For example, when x = 0, we have sinh 0 = (e⁰ – e⁰)/2 = 0. When x = 1, sinh 1 ≈ (2.718 – 0.3679)/2 ≈ 1.175.

例如，当 x = 0 时，sinh 0 = (e⁰ – e⁰)/2 = 0。当 x = 1 时，sinh 1 ≈ (2.718 – 0.3679)/2 ≈ 1.175。

3. Definition of cosh x | 双曲余弦的定义

The hyperbolic cosine is defined as cosh x = (eˣ + e⁻ˣ) / 2. It is an even function, meaning cosh(–x) = cosh x. The smallest value of cosh x is 1, which occurs at x = 0.

双曲余弦的定义为 cosh x = (eˣ + e⁻ˣ) / 2。它是一个偶函数，即 cosh(–x) = cosh x。cosh x 的最小值是 1，出现在 x = 0 处。

For instance, cosh 0 = (e⁰ + e⁰)/2 = 1. At x = 1, cosh 1 ≈ (2.718 + 0.3679)/2 ≈ 1.543.

例如，cosh 0 = (e⁰ + e⁰)/2 = 1。在 x = 1 时，cosh 1 ≈ (2.718 + 0.3679)/2 ≈ 1.543。

4. Definition of tanh x | 双曲正切的定义

The hyperbolic tangent is the ratio tanh x = sinh x / cosh x = (eˣ – e⁻ˣ) / (eˣ + e⁻ˣ). It is an odd function, and its values always lie between –1 and 1. As x grows large, tanh x approaches 1; as x becomes very negative, it approaches –1.

双曲正切定义为比值 tanh x = sinh x / cosh x = (eˣ – e⁻ˣ) / (eˣ + e⁻ˣ)。它是奇函数，函数值始终介于 –1 与 1 之间。当 x 很大时，tanh x 趋近 1；当 x 很负时，趋近 –1。

At x = 0, tanh 0 = 0/1 = 0. At x = 1, tanh 1 ≈ 1.175 / 1.543 ≈ 0.762.

在 x = 0 时，tanh 0 = 0/1 = 0。在 x = 1 时，tanh 1 ≈ 1.175 / 1.543 ≈ 0.762。

5. Graphs of sinh x and cosh x | sinh x 与 cosh x 的图像

The graph of y = sinh x looks like a smooth ‘S’ shape passing through the origin. It increases without bound for positive x and decreases without bound for negative x, showing point symmetry about the origin.

y = sinh x 的图像是一条通过原点的光滑 S 形曲线。当 x 为正时，曲线无限上升；当 x 为负时，无限下降，且关于原点呈点对称。

The graph of y = cosh x is shaped like a hanging chain (a catenary). It is symmetric about the y‑axis, with its lowest point at (0,1). For large |x|, the graph grows exponentially.

y = cosh x 的图像形似悬挂的链条（悬链线）。它关于 y 轴对称，最低点在 (0,1)。当 |x| 很大时，图像呈指数增长。

6. Graph of tanh x and Its Features | tanh x 的图像及其特征

The graph of y = tanh x is an S‑shaped curve that lies entirely between the horizontal lines y = –1 and y = 1. It crosses the origin and has two horizontal asymptotes: y = 1 and y = –1.

y = tanh x 的图像是一条完全位于水平线 y = –1 和 y = 1 之间的 S 形曲线。它通过原点，并有两条水平渐近线：y = 1 和 y = –1。

This shape is very useful in modelling processes that saturate, such as learning curves or population growth with limited resources.

这种形状在建模趋于饱和的过程中非常有用，比如学习曲线或资源有限的种群增长。

7. Fundamental Hyperbolic Identity | 基本双曲恒等式

Just as trigonometric functions have the identity sin² θ + cos² θ = 1, hyperbolic functions satisfy cosh² x – sinh² x = 1. You can verify this by substituting the exponential definitions.

就像三角函数有恒等式 sin² θ + cos² θ = 1 一样，双曲函数满足 cosh² x – sinh² x = 1。你可以通过代入指数定义来验证它。

Proof: cosh² x = [(eˣ + e⁻ˣ)/2]² = (e²ˣ + 2 + e⁻²ˣ)/4, sinh² x = (e²ˣ – 2 + e⁻²ˣ)/4. Subtracting gives (4)/4 = 1.

证明：cosh² x = [(eˣ + e⁻ˣ)/2]² = (e²ˣ + 2 + e⁻²ˣ)/4，sinh² x = (e²ˣ – 2 + e⁻²ˣ)/4。相减得 (4)/4 = 1。

8. Other Important Identities | 其他重要恒等式

Sinh and cosh have addition formulas similar to trigonometry but with some sign changes: sinh(x + y) = sinh x cosh y + cosh x sinh y; cosh(x + y) = cosh x cosh y + sinh x sinh y. For tanh, we have tanh(x + y) = (tanh x + tanh y) / (1 + tanh x tanh y).

sinh 和 cosh 有类似于三角函数的加法公式，但有一些符号变化：sinh(x + y) = sinh x cosh y + cosh x sinh y；cosh(x + y) = cosh x cosh y + sinh x sinh y。对于 tanh，有 tanh(x + y) = (tanh x + tanh y) / (1 + tanh x tanh y)。

Double‑argument formulas also follow: sinh 2x = 2 sinh x cosh x, and cosh 2x = cosh² x + sinh² x = 2 cosh² x – 1 = 2 sinh² x + 1.

倍角公式也类似：sinh 2x = 2 sinh x cosh x，cosh 2x = cosh² x + sinh² x = 2 cosh² x – 1 = 2 sinh² x + 1。

9. Relationship with Trigonometric Functions | 与三角函数的关系

There is a striking connection through complex numbers: sin(ix) = i sinh x, and cos(ix) = cosh x. Here i is the imaginary unit. This shows that hyperbolic functions are just trigonometric functions rotated in the complex plane.

通过复数有一种惊人的联系：sin(ix) = i sinh x，而 cos(ix) = cosh x。这里 i 是虚数单位。这表明双曲函数其实就是三角函数在复平面上的旋转。

For KS3, simply remember that removing the i’s from sine and cosine gives the hyperbolic counterparts. This link explains why their identities look so alike.

对 KS3 来说，只需记住从正弦和余弦中去掉 i 就能得到双曲版本。这种联系解释了为什么它们的恒等式看起来如此相似。

10. Real‑World Applications | 实际应用

Hyperbolic functions model the shape of a freely hanging chain or cable – the catenary. The equation of a catenary is y = a cosh(x/a). They also describe the velocity of a wave in deep water and appear in Einstein’s special relativity for adding velocities.

双曲函数可模拟自由悬挂的链条或电缆的形状——悬链线。悬链线的方程是 y = a cosh(x/a)。它们还描述深水波的波速，并出现在爱因斯坦狭义相对论的速度相加公式中。

In engineering, tanh is used in heat transfer and chemical reaction rates. Even the famous Gateway Arch in St. Louis is an inverted catenary, designed using cosh.

在工程中，tanh 用于传热和化学反应速率。就连著名的圣路易斯拱门也是一个倒置的悬链线，是用 cosh 设计的。

11. Key Values to Remember | 需要记住的关键值

Just as you memorise sin 30° = ½, for hyperbolic functions you can memorise a few exact values:

sinh 0 = 0, cosh 0 = 1, tanh 0 = 0
sinh(ln φ) = ½, where φ is the golden ratio (1 + √5)/2.

Most other values are approximated, but these highlight important behaviour.

就像你记住 sin 30° = ½ 一样，对于双曲函数，你可以记住一些精确值：

sinh 0 = 0，cosh 0 = 1，tanh 0 = 0
sinh(ln φ) = ½，其中 φ 是黄金比例 (1 + √5)/2。

大多数其他的值是近似的，但这些能突出重要行为。

12. Summary and Next Steps | 总结与展望

Hyperbolic functions sinh, cosh, and tanh are defined from exponentials, drawn from a hyperbola. They have elegant identities, interesting graphs, and wide applications. At KS3, focus on understanding the definitions and their basic shapes – this will make A‑level study much smoother.

双曲函数 sinh、cosh 和 tanh 由指数定义，源自双曲线。它们有优雅的恒等式、有趣的图像和广泛的应用。在 KS3 阶段，集中理解定义及其基本形状——这会让 A‑level 的学习顺畅很多。

Try plotting y = sinh x, y = cosh x, and y = tanh x using a graphing tool, and verify cosh² x – sinh² x = 1 for a few values of x. Explore the catenary by holding a chain at both ends and observing its curve – it’s a real‑life cosh!

尝试用绘图工具画出 y = sinh x、y = cosh x 和 y = tanh x 的图像，并针对几个 x 值验证 cosh² x – sinh² x = 1。用手握住链条两端观察它的曲线——那就是现实中的 cosh！

Published by TutorHao | 数学 Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Hyperbolic Functions for KS3 | KS3 数学：双曲函数 考点精讲