Hyperbolic sine double-angle calculator
Evaluate sinh(2x) and see where it comes from: at 0 the result is 0, because sinh(0) is 0 and cosh(0) is 1. Both factors are shown, and values too large to represent are refused instead of shown as infinity. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- X
- 0.5
- Sinh x
- 0.5210953055
- Cosh x
- 1.1276259652
- Sinh double x
- 1.1752011936
The hyperbolic sine double-angle calculator evaluates sinh(2x) from one real argument by applying the identity sinh(2x) = 2 sinh(x) cosh(x). It returns the normalized input, both hyperbolic function values used in the product, and the final numeric result. This makes the calculation useful not only for obtaining an answer, but also for checking algebra and tracing intermediate values. Inputs must be finite, and calculations that exceed the finite numeric range are rejected instead of returning an unusable infinity.
Enter the real argument x
Provide one finite real number in the x field. The argument is dimensionless: there is no degrees-versus-radians choice because hyperbolic functions are defined through exponential expressions rather than circular angle measurement. Positive, negative, fractional, and zero values are supported.
After validation, the calculator normalizes the supplied value to a JavaScript number and reports it as x in the response. For a negative argument, sinh(x) is negative while cosh(x) remains positive, so the final result is negative. These basic cases provide quick checks on the sign and symmetry of a result before using it elsewhere.
Follow the double-angle identity
The solver calculates sinh(x) and cosh(x) first, then multiplies them according to sinh(2x) = 2 sinh(x) cosh(x). This is a hyperbolic identity, not the ordinary sine identity sin(2x) = 2 sin(x) cos(x), even though their written structures look similar. Hyperbolic sine can be defined as (e^x - e^-x) / 2, while hyperbolic cosine is (e^x + e^-x) / 2.
Multiplying those definitions and doubling the product yields (e^(2x) - e^(-2x)) / 2, which is precisely sinh(2x). Floating-point arithmetic can produce the usual small representation differences found in numerical software, so results should be interpreted at an appropriate precision for the surrounding application. The implementation does not introduce arbitrary rounding; preserving the native numeric result lets callers choose their own display precision, tolerance, or significant-figure policy without losing information prematurely.
Interpret finite results and range errors
The factor cosh(x) is essential and generally makes those two quantities different. Because hyperbolic functions grow exponentially in magnitude, sufficiently large positive or negative arguments exceed the finite range of standard double-precision numbers. For ordinary finite inputs within range, execution is deterministic and requires no network service, stored state, random value, or current time.
Use the intermediate fields to audit a symbolic derivation, compare a numerical library, or feed a transparent result into another computation. If your application expects very large magnitudes, consider whether logarithmic representations or an arbitrary-precision mathematics system would better match that requirement.
What you can do with it
Check a hyperbolic identity exercise
Compare a handwritten evaluation of 2 sinh(x) cosh(x) with the returned intermediate factors and final result.
Validate numerical software
Use deterministic reference calculations to test a sinh double-angle implementation within normal floating-point range.
Build a transparent calculation step
Include sinh(x), cosh(x), and sinh(2x) in a larger engineering or mathematical workflow without hiding the identity used.
FAQ
What formula does the calculator use?
It uses the identity sinh(2x) = 2 sinh(x) cosh(x).
Is x measured in degrees or radians?
Neither choice is required. Hyperbolic functions take a dimensionless real argument rather than an angle-unit setting.
Can x be negative or zero?
Yes. Any finite real number is accepted as long as the intermediate values and result remain finite.
Why can a large finite input return an error?
Hyperbolic functions grow exponentially, so a large argument can exceed the finite range of standard double-precision arithmetic.
Is sinh(2x) the same as twice sinh(x)?
No. The identity multiplies twice sinh(x) by cosh(x), and that extra factor generally makes the two results differ. Only at zero do both give the same value, zero, since cosh(0) is one.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/trig/sinh-double-angle \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"x":0.5}'const res = await fetch("https://api.kit.forhosting.com/trig/sinh-double-angle", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"x": 0.5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/trig/sinh-double-angle",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"x": 0.5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/trig/sinh-double-angle", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"x":0.5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"x":0.5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/trig/sinh-double-angle", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"x": 0.5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "trig.sinh_double_angle",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Enter the real argument x
The declared argument and value aliases are also accepted by the API, although x is the clearest field name for formulas and generated client types. Blank text, nonnumeric text, NaN, and positive or negative infinity are rejected with an invalid_input error. This strict behavior matters in automated calculations, where silently propagating a non-finite value could contaminate later steps. For x = 0, both sides of the identity give zero: sinh(0) is zero, cosh(0) is one, and their doubled product is zero.
Follow the double-angle identity
The response exposes sinh_x and cosh_x so you can inspect the two factors rather than receiving an unexplained final value. It also includes a formula label that states the operation performed.
Interpret finite results and range errors
Read sinh_double_x as sinh(2x), not as 2 sinh(x). The calculator checks the intermediate sinh(x), cosh(x), and final product, and reports invalid_input when any required value is not finite. That response is more reliable than returning Infinity, which is not valid JSON and is rarely useful in a downstream numeric workflow. The same pure solver supports API and browser execution, so a given input follows the same calculation path in either channel. Automated API calls use the published base price of $0.002 per request, while the browser experience is suitable for quick interactive checks.
Technical questions
What does an API calculation cost?
Each API request uses the published base price of $0.002. The calculation can also run in the browser.