Sinh
Calculate sinh(x) for any real number using the exponential formula. Free online hyperbolic sine calculator with instant results and interactive graph.
About This Calculator
The hyperbolic sine function, denoted sinh(x), is a fundamental hyperbolic function defined as sinh(x) = (e^x - e^(-x)) / 2. It is the odd counterpart to hyperbolic cosine (cosh) and models many natural phenomena involving exponential growth and decay.
Unlike regular trigonometric sine which is periodic and bounded, hyperbolic sine grows exponentially as x increases and decreases exponentially (negatively) as x decreases. The function passes through the origin (sinh(0) = 0) and is symmetric about the origin: sinh(-x) = -sinh(x).
Hyperbolic sine is essential in physics and engineering: it describes the shape of a hanging cable (catenary curve) when combined with cosh, appears in solutions to the wave equation, models thermal diffusion, and is used in special relativity for Lorentz transformation calculations.
Frequently Asked Questions
What is hyperbolic sine (sinh)?
Hyperbolic sine (sinh) is a hyperbolic function defined as sinh(x) = (e^x - e^(-x)) / 2. Unlike regular sine which is periodic, sinh grows exponentially for large positive x and approaches negative exponential for large negative x.
How does sinh differ from regular sine?
Regular sine (sin) is periodic with period 2pi and ranges between -1 and 1. Hyperbolic sine (sinh) is not periodic and grows without bound. While sin uses the unit circle, sinh describes a catenary curve and relates to hyperbolas.
What is the formula for sinh?
The exponential formula for hyperbolic sine is: sinh(x) = (e^x - e^(-x)) / 2. Some common values: sinh(0) = 0, sinh(1) ≈ 1.1752, sinh(-1) ≈ -1.1752. It is an odd function: sinh(-x) = -sinh(x).
Where is hyperbolic sine used in real life?
Hyperbolic sine appears in physics (catenary curves for hanging cables and power lines), in engineering (calculating the shape of suspension bridges and arch supports), in special relativity (Lorentz transformations), and in mathematics (solving differential equations).
How does sinh relate to other hyperbolic functions?
Hyperbolic sine relates to other hyperbolic functions through identities: tanh(x) = sinh(x) / cosh(x), cosh^2(x) - sinh^2(x) = 1 (the hyperbolic Pythagorean identity), and d/dx sinh(x) = cosh(x). These mirror regular trig identities with sign changes.
Is this sinh calculator free?
Yes, all calculators on Calculy are completely free to use. No registration or download required.
What is the derivative of sinh?
The derivative of sinh(x) is cosh(x), the hyperbolic cosine. This mirrors how the derivative of sin(x) is cos(x) in regular trigonometry. Specifically, d/dx sinh(x) = cosh(x) = (e^x + e^(-x))/2.
How do you calculate sinh on a basic calculator?
To calculate sinh(x) on a basic calculator, first compute e^x and store it, then compute e^(-x) and store it, subtract e^(-x) from e^x, and divide the result by 2. Alternatively, use the exponential function key if available: (e^x - e^(-x)) / 2.