Hyperbolic Functions Calculator
Compute sinh, cosh, tanh, coth, sech, and csch for x in radians or degrees, examine definitions, and review cosh²x − sinh²x.
Note
Cosh²x − sinh²x should equal 1; deviations arise from floating-point rounding.
Notes
| Function | Value | Definition |
|---|---|---|
| sinh x | — | \(\dfrac{e^x - e^{-x}}{2}\) |
| cosh x | — | \(\dfrac{e^x + e^{-x}}{2}\) |
| tanh x | — | \(\dfrac{\sinh x}{\cosh x}\) |
| coth x | — | \(\dfrac{\cosh x}{\sinh x}\) |
| sech x | — | \(\dfrac{1}{\cosh x}\) |
| csch x | — | \(\dfrac{1}{\sinh x}\) |
How to use
Enter x, choose radians or degrees, select the display format, and press Calculate to evaluate all six hyperbolic functions at once.
Methodology
The calculator convert degrees to radians when needed, uses exponential definitions to compute each value, and reports deviations from the identity cosh²x − sinh²x = 1 to highlight numerical precision.
Full original guide (expanded)
Definitions
Identity cosh²x − sinh²x = 1
This parallels the circular identity \(\cos^2x + \sin^2x = 1\) but with a subtraction sign because of the hyperbola geometry.
Precision, domains, and notes
For large |x|, exponentials may overflow and the identity check will drift from 1. When sinh x = 0, coth and csch are undefined; when cosh x = 0 they are undefined as well, but cosh x never actually vanishes for real x.
The calculator displays both radian and degree representations so you can compare the hyperbolic argument with typical angle measures.
Applications
Hyperbolic functions describe hanging cables, certain differential equation solutions, rapidity in relativity, and appear whenever exponential growth must be balanced symmetrically for positive and negative direction.