Hyperbolic cotangent calculator
The hyperbolic cotangent of 1 is 1.3130352854993312, the ratio of a cosh of 1.5430806348152437 to a sinh of 1.1752011936438014, and all three values appear together. Zero is refused because sinh is zero there, so no invented value slips through. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Value
- 1
- Cosh
- 1.5430806348
- Sinh
- 1.1752011936
- Coth
- 1.3130352855
The hyperbolic cotangent calculator evaluates coth(x) for a finite, nonzero real number by forming the ratio of hyperbolic cosine to hyperbolic sine. It returns the input, the component values, the resulting ratio, and the formula used, making the calculation easy to inspect or automate. Zero is rejected because sinh(0) equals zero, so the ratio has no defined real value there.
What hyperbolic cotangent means
Hyperbolic cotangent, written coth(x), belongs to the family of hyperbolic functions used to describe exponential growth, catenary curves, differential equations, heat transfer, fluid models, and several probability distributions. Its definition is straightforward: divide cosh(x) by sinh(x). Although the name resembles ordinary cotangent, coth does not take an angle on a circle.
It operates on a real number through exponential relationships, with cosh(x) equal to the average of exp(x) and exp(−x), and sinh(x) equal to half their difference. For a positive input, the result is greater than one and approaches one as the input grows. For a negative input, the result is less than negative one and approaches negative one as the magnitude grows.
The function is odd, so coth(−x) equals −coth(x). This calculator exposes the component values alongside the quotient, helping you verify a textbook derivation, inspect an intermediate scientific calculation, or compare a software library result with the defining formula rather than treating coth as an unexplained black box.
How the calculator handles the domain and large values
Enter one finite real value. Decimal notation and scientific notation are accepted by the solver, while missing values, nonnumeric text, NaN, and positive or negative infinity are rejected as invalid input. Zero receives special treatment because sinh(0) is exactly zero.
For ordinary magnitudes, the engine directly evaluates Math.cosh(x) and Math.sinh(x), then divides the former by the latter. Very large magnitudes need an additional numerical safeguard because both component functions can overflow the JavaScript number range even though their ratio is close to one or negative one. In that region, the solver uses the algebraically equivalent scaled ratio sign(x) times (1 + exp(−2|x|)) divided by (1 − exp(−2|x|)).
The response marks the raw components as overflow while preserving a finite, mathematically meaningful coth result.
Reading and using the result
The response includes value, cosh, sinh, coth, and a formula label. The coth field is the primary answer. The cosh and sinh fields make it possible to audit the quotient for inputs whose components fit within the finite number range.
Near zero, the magnitude of coth grows rapidly, which is expected: sinh(x) is approximately x for small x while cosh(x) is approximately one, so their ratio behaves approximately like 1/x. Keep that sensitivity in mind when an upstream measurement may round a tiny positive or negative quantity to zero.
What you can do with it
Check a differential-equation step
Evaluate coth at a boundary value when verifying an analytic solution involving hyperbolic functions.
Validate scientific software
Compare a library or spreadsheet result with the explicit cosh-over-sinh definition and its component values.
Automate engineering calculations
Insert deterministic hyperbolic cotangent values into heat-transfer, wave, or transport calculation pipelines.
FAQ
What formula does the calculator use?
It uses coth(x) = cosh(x) / sinh(x), with an algebraically equivalent scaled form at very large magnitudes to prevent component overflow.
Why is coth(0) undefined?
Because sinh(0) equals zero while cosh(0) equals one, making the defining ratio a division by zero.
Can I enter a negative number?
Yes. Any finite nonzero real number is valid, and coth is odd, so a negative input produces the negation of the corresponding positive result.
Why do large inputs show overflow for cosh and sinh?
Those components can exceed the JavaScript finite number range. The solver still computes their finite ratio with a scaled equivalent of the same formula.
What does one calculation cost?
The same deterministic solver can run free in the browser on this page.
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/coth \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"value":1}'const res = await fetch("https://api.kit.forhosting.com/trig/coth", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"value": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/trig/coth",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"value": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/trig/coth", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"value":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"value":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/trig/coth", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"value": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "trig.coth",
"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
The solver is deterministic, uses no network service, and shares the same pure calculation between browser and API execution.
How the calculator handles the domain and large values
Dividing cosh(0), which equals one, by that zero denominator would not produce a defined real number, so the request fails with a clear invalid_input message instead of returning an invented value or a misleading infinity.
Reading and using the result
For extremely large magnitudes, those two component fields use explicit overflow text because JSON cannot safely carry JavaScript Infinity, while coth remains a finite numeric result obtained from the scaled equivalent. Each API request processes one value as one item and costs $0.002; the page can also run the same pure solver in the browser. Identical inputs produce identical JSON without network calls, random values, stored state, or clock access, making the capability suitable for reproducible worksheets, automated tests, numerical pipelines, and scientific application checks.