Hyperbolic sine calculator
Enter a number x and get sinh(x), with the formula it comes from.
Decimals may use a point or a comma. A sum works too, such as 2^10 or sqrt(2).
Fill in the fields; the answer appears here straight away.
How it's worked out
How it works
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Enter the values
Type the numbers or the formula. Decimals may use a point or a comma. No idea? Click Fill in an example.
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Instant answer
The answer appears as you type, with the most important intermediate values.
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See the working
Under "How it's worked out" you see the steps, handy for checking your own calculation.
Definition
sinh(x) = (eˣ − e⁻ˣ) / 2. Unlike the ordinary sine it doesn't repeat: it grows without limit in both directions. sinh(0) = 0 and sinh(−x) = −sinh(x).
Where it appears
Together with cosh it describes a hanging chain or cable (a catenary), and it shows up in special relativity and in solutions of differential equations. The identity cosh²(x) − sinh²(x) = 1 plays the role of sin² + cos² = 1. See also cosh and tanh.
Values to compare with
| x | sinh(x) |
|---|---|
| 0 | 0 |
| 0.5 | 0.5211 |
| 1 | 1.1752 |
| 2 | 3.6269 |
| 5 | 74.2032 |
Small and large x
For small x, sinh(x) is almost x: sinh(0.1) ≈ 0.1002. For large x the term e⁻ˣ vanishes and sinh(x) ≈ eˣ ÷ 2, which grows as fast as the exponential. The number e and powers of it are found with the exponent calculator.
Connection with cosh
Add the two: cosh(x) + sinh(x) = eˣ, and subtract them: cosh(x) − sinh(x) = e⁻ˣ. That is why the identity cosh² − sinh² = 1 holds. For the circular functions see the trigonometry calculator.
Frequently asked questions
Is sinh the same as sin?
No. sin(x) repeats and stays between −1 and 1, while sinh(x) grows without limit. They are related through complex numbers, but they are different functions.
Why is it called hyperbolic?
The point (cosh t, sinh t) lies on the hyperbola x² − y² = 1, just as (cos t, sin t) lies on the unit circle.
What is sinh of a negative number?
The negative of sinh of the positive number: sinh(−x) = −sinh(x). It is an odd function, so sinh(−1) ≈ −1.1752.
What is sinh(0)?
Zero. (e⁰ − e⁰) ÷ 2 = 0.