Hyperbolic functions

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:

cosh and sinh are the average and half-difference of e^x and e^-x

Key identities

• Main identity 恒等式: $\cosh^2 x - \sinh^2 x = 1$ (compare with $\cos^2 x + \sin^2 x = 1$).
• Addition: $\sinh(A + B) = \sinh A \cosh B + \cosh A \sinh B$.

A hanging cable forms a catenary, the curve of the hyperbolic cosine y = a cosh(x/a)

Derivatives 导数

• $\dfrac{d}{dx}\sinh x = \cosh x$, $\quad\dfrac{d}{dx}\cosh x = \sinh x$ (no minus sign!).
• $\dfrac{d}{dx}\tanh x = \text{sech}^2 x$.

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

Write from cosh of ln 2

• Find cosh of ln 2 from the exponential definition.
• The identity is cosh squared minus sinh squared, not plus.