What is the formula for sinh(x)?

sinh(x) = (e^x - e^(-x)) / 2, where e is Euler's number (~2.71828). It is the hyperbolic analogue of the sine function.

How is sinh different from sin?

Regular sine is defined on a unit circle. Hyperbolic sine is defined via exponentials and is related to a unit hyperbola. sinh(x) is unbounded as x grows, whereas sin(x) stays within [-1, 1].

Is sinh(x) always defined?

Yes. sinh(x) is defined for all real numbers. It is an odd function, meaning sinh(-x) = -sinh(x), and sinh(0) = 0.

What are typical uses of sinh in engineering?

The shape of a hanging cable (catenary) is described by cosh and sinh. They also appear in heat transfer, fluid mechanics, and special relativity formulas.

Can I compute sinh for very large numbers?

Yes, though for very large inputs (above ~709 in magnitude) floating-point overflow may occur. The calculator flags this and displays the closest representable value.

Hyperbolic Sine Calculator

Enter any real number and instantly compute its hyperbolic sine (sinh) with a step-by-step formula breakdown.

Input

Value of x

Accepts any real number. Decimals and negatives are supported.

Quick Examples

Result

Enter a value and press Calculate.

Formula Reference

Definition

sinh(x) = (e^x - e^(-x)) / 2

Odd Function

sinh(-x) = -sinh(x)

Identity

cosh²(x) - sinh²(x) = 1

Copied!

Summary

The hyperbolic sine function, sinh(x), is defined as (e^x - e^(-x)) / 2. It appears in engineering, physics, and mathematics — particularly in catenary curves, wave equations, and solutions to differential equations. This calculator computes sinh(x) for any real number and shows each step of the exponential formula so you can verify your work.

How it works

Enter any real number x in the input field.
The calculator evaluates e^x and e^(-x) using the natural exponential.
It then subtracts the two values: e^x minus e^(-x).
The result is divided by 2 to yield sinh(x).
All intermediate values are displayed so you can follow every step.

Use cases

Verify homework or textbook answers for hyperbolic function problems.
Check sinh values when studying catenary cable or arch shapes.
Confirm numerical solutions while solving differential equations.
Explore how sinh(x) grows compared to sin(x) for large inputs.
Convert between hyperbolic and exponential forms for exam preparation.
Quick reference during physics or engineering calculations.

Frequently Asked Questions

Last updated: 2026-05-29 · Reviewed by Nham Vu