Hyperbolic functions
| English | Chinese | Pinyin |
|---|---|---|
| catenary | 悬链线 | xuán liàn xiàn |
| hyperbolic function | 双曲函数 | shuāng qū hán shù |
| exponential | 指数 | zhǐ shù |
| identity | 恒等式 | héng děng shì |
| derivative | 导数 | dǎo shù |
| inverse | 反函数 | fǎn hán shù |
The curves that hang from chains
- A hanging chain forms a catenary 悬链线 curve — not a parabola, but the graph of $\cosh x$.
- Hyperbolic functions 双曲函数 are built from exponentials 指数 and appear everywhere: suspension bridges, power lines, and even the shape of the universe in general relativity.
Definitions
- Built from the exponential function:
Worked example. $\cosh 0 = \dfrac{e^0 + e^0}{2} = \dfrac{2}{2} = 1$. $\sinh 0 = \dfrac{1 - 1}{2} = 0$. So the catenary passes through $(0, 1)$, not the origin.
cosh and sinh are the average and half-difference of e^x and e^-x
Hyperbolic functions
y = a cosh(x)
cosh is the catenary (a hanging chain) — even, with its minimum at (0, a).
Using cosh x = (eˣ + e⁻ˣ)/2, what is cosh 0?
cosh 0 = (e⁰ + e⁰)/2 = (1 + 1)/2 = 1.
Using sinh x = (eˣ − e⁻ˣ)/2, what is sinh 0?
sinh 0 = (1 − 1)/2 = 0.
Key identities
- Main identity 恒等式: $\cosh^2 x - \sinh^2 x = 1$ (compare with $\cos^2 x + \sin^2 x = 1$).
- Addition: $\sinh(A + B) = \sinh A \cosh B + \cosh A \sinh B$.
Minus sign, not plus. The identity is $\cosh^2 x - \sinh^2 x = 1$ (minus), not $\cosh^2 x + \sinh^2 x = 1$. This is the key difference from trigonometric identities.
A hanging cable forms a catenary, the curve of the hyperbolic cosine y = a cosh(x/a)
For any x, what does cosh²x − sinh²x equal?
The hyperbolic identity is cosh²x − sinh²x = 1.
The identity cosh²x + sinh²x = 1 is correct.
The correct identity is cosh²x − sinh²x = 1 (minus, not plus).
Derivatives 导数
- $\dfrac{d}{dx}\sinh x = \cosh x$, $\quad\dfrac{d}{dx}\cosh x = \sinh x$ (no minus sign!).
- $\dfrac{d}{dx}\tanh x = \text{sech}^2 x$.
The derivative of sinh x is:
d/dx(sinh x) = cosh x (no minus sign, unlike the trig derivative d/dx(sin x) = cos x).
The derivative of cosh x is:
d/dx(cosh x) = sinh x (no minus sign, unlike d/dx(cos x) = −sin x).
Inverse 反函数 hyperbolic functions
- Inverse forms are logarithmic:
- $\sinh^{-1}x = \ln(x + \sqrt{x^2 + 1})$
- $\cosh^{-1}x = \ln(x + \sqrt{x^2 - 1})$ for $x \geq 1$.
You've got it
- $\cosh x = \dfrac{e^x + e^{-x}}{2}$, $\sinh x = \dfrac{e^x - e^{-x}}{2}$
- key identity: $\cosh^2 x - \sinh^2 x = 1$
- inverse hyperbolics have logarithmic forms