Sinh Calculator
Calculate hyperbolic sine (sinh) and inverse hyperbolic sine (arsinh) with step-by-step solutions
Hyperbolic Sine Calculator
Any real number
Results
Step-by-step Calculation
Common Hyperbolic Sine Values
| x | sinh(x) | cosh(x) | tanh(x) |
|---|---|---|---|
| 0 | 0.0000 | 1.0000 | 0.0000 |
| 0.5 | 0.5211 | 1.1276 | 0.4621 |
| 1 | 1.1752 | 1.5431 | 0.7616 |
| 1.5 | 2.1293 | 2.3524 | 0.9051 |
| 2 | 3.6269 | 3.7622 | 0.9640 |
| -0.5 | -0.5211 | 1.1276 | -0.4621 |
| -1 | -1.1752 | 1.5431 | -0.7616 |
| -2 | -3.6269 | 3.7622 | -0.9640 |
Properties of sinh(x)
Odd Function
sinh(-x) = -sinh(x)
Increasing
Always increasing for all x
Zero at Origin
sinh(0) = 0
Unbounded
Range: (-∞, ∞)
Key Formulas
Quick Tips
sinh grows exponentially for large positive x
sinh approaches -∞ for large negative x
Used in catenary curves and wave equations
Related to exponential functions and logarithms
Understanding Hyperbolic Sine
What is Hyperbolic Sine?
The hyperbolic sine function (sinh) is a hyperbolic function that's analogous to the ordinary sine function. It's defined using exponential functions and appears in many areas of mathematics, physics, and engineering.
Mathematical Definition
sinh(x) = (e^x - e^(-x)) / 2
This definition shows that sinh is built from exponential functions, which explains its growth behavior and many of its properties.
Key Properties
- •Domain: All real numbers (-∞, ∞)
- •Range: All real numbers (-∞, ∞)
- •Symmetry: Odd function (point symmetric about origin)
- •Monotonicity: Strictly increasing everywhere
- •Derivative: d/dx sinh(x) = cosh(x)
Connection to Hyperbola: Points (cosh(t), sinh(t)) trace out a hyperbola, just as (cos(t), sin(t)) trace out a circle.
Applications
Physics
Wave equations, relativity, quantum mechanics, electromagnetic fields
Engineering
Catenary curves, hanging cables, heat transfer, signal processing
Mathematics
Complex analysis, differential equations, integration techniques