What Is the Hyperbolic Functions (sinh, cosh, tanh) Calculator?
Hyperbolic functions are mathematical functions defined through combinations of the exponential function eˣ.
Just as points (cos θ, sin θ) trace out a unit circle x² + y² = 1, points (cosh t, sinh t) trace out the right branch of the unit hyperbola x² - y² = 1.
How Does the Hyperbolic Functions (sinh, cosh, tanh) Calculator Work?
sinh(x) = (eˣ - e⁻ˣ) / 2.
cosh(x) = (eˣ + e⁻ˣ) / 2.
tanh(x) = sinh(x) / cosh(x) = (e²ˣ - 1) / (e²ˣ + 1).
Reciprocals sech(x) = 1/cosh(x), csch(x) = 1/sinh(x), and coth(x) = 1/tanh(x) are also computed.
Hyperbolic Functions (sinh, cosh, tanh) Calculator Formula & Variables
The core mathematical equation utilized by this calculator is expressed as:
Hyperbolic functions are analogs of ordinary circular trigonometric functions defined using the natural exponential function eˣ.
How to Use the Hyperbolic Functions (sinh, cosh, tanh) Calculator
- Enter input argument x.
- View numerical outputs for sinh(x), cosh(x), tanh(x), and sech(x).
- Verify the fundamental identity cosh²(x) - sinh²(x) = 1.
Step-by-Step Example Calculation
x = 1.0
Input Values:
Understanding Your Result
cosh(x) is an even function with a minimum value of 1 at x = 0.
sinh(x) is an odd function crossing 0 at x = 0.
tanh(x) strictly approaches +1 as x → +∞ and -1 as x → -∞.
Factors That Affect the Result
- Magnitude: Hyperbolic cosine grows exponentially with large positive or negative arguments.
When Should You Use This Calculator?
- Modeling catenary suspension bridges and sagging overhead electrical transmission lines: y = a · cosh(x/a).
- Special relativity rapidities and Lorentz transformation boosts.
Assumptions & Limitations
- Evaluates real numbers; complex arguments are not supported.
Frequently Asked Questions
Calculation Accuracy & Reference Note
Implements standard IEEE 754 Math.sinh and Math.cosh routines.