📐 Hyperbolic Geometry & Catenary Mechanics
Hyperbolic Functions & Inverse Calculator
Evaluate direct hyperbolic functions ($\sinh x, \cosh x, \tanh x, \operatorname{sech} x$) and inverse hyperbolic functions ($\operatorname{arsinh}, \operatorname{arcosh}, \operatorname{artanh}$) across real arguments.
Input Argument ($x$)
x = 0
x = 0.5
x = 1.0
x = 2.0
x = 3.0
Primary Hyperbolic Values for x = 1.0
sinh = 1.1752, cosh = 1.5431
$\sinh(x)$
1.1752
$\cosh(x)$
1.5431
$\tanh(x)$
0.7616
💡 Hyperbolic Pythagorean Identity:
$$\cosh^2(1.0) - \sinh^2(1.0) = (1.5431)^2 - (1.1752)^2 = 2.3811 - 1.3811 = 1.0000$$
Hyperbolic Function Definitions
Analogous to circular trigonometric functions on the unit circle ($x^2 + y^2 = 1$), hyperbolic functions parametrize the unit hyperbola ($x^2 - y^2 = 1$):
- Hyperbolic Sine: $$\sinh(x) = \frac{e^x - e^{-x}}{2}$$
- Hyperbolic Cosine: $$\cosh(x) = \frac{e^x + e^{-x}}{2}$$
- Hyperbolic Tangent: $$\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^{2x} - 1}{e^{2x} + 1}$$
- Catenary Curve: A hanging flexible cable under uniform gravity forms $y = a \cosh\left(\frac{x}{a}\right)$.