Hyperbolic functions
Hyperbolic functions
- Built from the exponential function:
$$\cosh x = \frac{e^x + e^{-x}}{2}, \qquad \sinh x = \frac{e^x - e^{-x}}{2}, \qquad \tanh x = \frac{\sinh x}{\cosh x}$$
- Main identity: $\cosh^2 x - \sinh^2 x = 1$.
- Inverse forms are logarithmic, e.g. $\sinh^{-1}x = \ln(x + \sqrt{x^2 + 1})$.
Practice
Using cosh x = (eˣ + e⁻ˣ)/2, what is cosh 0?
cosh 0 = (e⁰ + e⁰)/2 = (1 + 1)/2 = 1.
Practice
For any x, what does cosh²x − sinh²x equal?
The hyperbolic identity is cosh²x − sinh²x = 1.
Practice
Using sinh x = (eˣ − e⁻ˣ)/2, what is sinh 0?
sinh 0 = (1 − 1)/2 = 0.
You've got it
Key idea
- $\cosh x = \dfrac{e^x + e^{-x}}{2}$, $\sinh x = \dfrac{e^x - e^{-x}}{2}$
- key identity: $\cosh^2 x - \sinh^2 x = 1$
- inverse hyperbolics have logarithmic forms