Hyperbolic Functions Calculator
Calculate sinh(x), cosh(x), tanh(x) and their reciprocals — csch(x), sech(x), coth(x) — for any real number x, the hyperbolic analogues of the circular trig functions.
Input
Output
| Function | Value |
|---|---|
| No data yet | |
More ways to use this tool
REST API
curl -X POST https://api.iotools.cloud/v1/tool/hyperbolic-functions-calculator \
-H "Authorization: Bearer YOUR_API_KEY" \
-H "Content-Type: application/json" \
-d '{
"x": "0"
}'Swap in your own key from your account. The tool's fields are the body — no wrapper.
Ask an AI agent
Use the IOTools `hyperbolic-functions-calculator` tool (Hyperbolic Functions Calculator) on this input:
YOUR_INPUT_HEREPaste this at any agent connected to the IOTools MCP server, then add your input.
Embed widget
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src="https://iotools.cloud/embed/hyperbolic-functions-calculator/"
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title="Hyperbolic Functions Calculator — iotools.cloud"
sandbox="allow-scripts allow-forms allow-same-origin allow-downloads allow-popups allow-popups-to-escape-sandbox"
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<script src="https://iotools.cloud/embed.js" async></script>Drop this into your own page — free, no key required, just a link back.
| Cost per API/MCP call | From 5 credits |
|---|---|
| Need more credits? | View pricing |
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Guides
What does this calculator do?
Enter a real number x and this tool computes all six hyperbolic functions at that point: sinh(x), cosh(x), tanh(x), and their reciprocals csch(x), sech(x), and coth(x). Hyperbolic functions are the analogues of the ordinary (circular) trig functions, but built from e^x instead of points on a unit circle — they're defined using a hyperbola instead.
How to use it
- Enter any real number for x — positive, negative, or zero.
- Read all six function values from the result table.
csch(x) and coth(x) are undefined at x = 0, since both divide by sinh(0) = 0 — the calculator flags those cases explicitly instead of showing an error.
The math
The three primary hyperbolic functions are defined directly from the exponential function:
sinh(x) = (eˣ − e⁻ˣ) / 2
cosh(x) = (eˣ + e⁻ˣ) / 2
tanh(x) = sinh(x) / cosh(x)The other three are just reciprocals of these:
csch(x) = 1 / sinh(x)
sech(x) = 1 / cosh(x)
coth(x) = 1 / tanh(x)Unlike the circular functions, sinh and tanh are odd (sinh(−x) = −sinh(x)) while cosh is even (cosh(−x) = cosh(x)). cosh(x) is always ≥ 1, and tanh(x) is always strictly between −1 and 1, approaching those bounds as x → ±∞. sinh and cosh grow very quickly — past roughly |x| = 710 they exceed what a standard 64-bit floating-point number can represent, and the calculator reports that as an overflow rather than a wrong number.
Example use cases
- Catenary curves — the shape a hanging chain or cable naturally forms is described by
y = a·cosh(x/a), used in bridge and power-line engineering. - Special relativity — rapidity (a velocity-like quantity that adds linearly under Lorentz boosts) is defined using
tanhandartanh. - Differential equations — hyperbolic functions are the natural solutions to equations like
y'' = y, showing up throughout physics and engineering. - Neural networks —
tanhis a classic activation function, valued for being smooth and bounded between −1 and 1.
Privacy
All calculations run locally in your browser — your input is never sent to a server.
Related tools
For another special-function evaluation, see the Gamma Function Calculator.