Arcsecant in degrees calculator
The arcsecant of 2 is 60° and that of −2 is 120°. Enter any number of 1 or more, or of −1 or less, and get in degrees the principal angle whose secant it is, between 0 and 180. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Value
- 2
- Degrees
- 60
- Radians
- 1.0471975512
- Unit degrees
- deg
- Unit radians
- rad
When you know a secant value and need the angle that produced it, the inverse secant—also written arcsec or asec—recovers the principal angle without drawing or iteration. This calculator accepts one finite number whose absolute value is at least one, applies the principal arcsecant via the identity arcsec of x equals arccos of one over x, and returns the result primarily in degrees so classroom work and engineering pipelines share one stable contract.
How inverse secant recovers the principal angle from a secant
The secant function is the reciprocal of cosine: for any angle where cosine is nonzero, sec of theta equals one over cos of theta. Because cosine never leaves the closed interval from negative one to one, the real range of secant is everything outside the open unit interval—exactly the set of real numbers with absolute value at least one. In degrees that same principal range is zero through one hundred eighty excluding ninety.
Computationally the service uses the stable identity arcsec of x equals acos of one over x, then scales radians by one hundred eighty over pi. Edge cases are textbook checks: arcsec of one is zero degrees, arcsec of negative one is one hundred eighty degrees, arcsec of two is sixty degrees, and arcsec of negative two is one hundred twenty degrees.
Because the path is a single reciprocal, one library arccosine, and a linear unit conversion, it is fully deterministic: the same secant always produces the same pair of angles within floating-point rounding.
Where inverse secant shows up in real work
Secondary-school and college trigonometry courses often introduce reciprocal inverse functions after arccos and arcsin; a trusted calculator with a strict domain check speeds homework grading and removes spreadsheet mistakes where someone applies arcsec to a ratio inside the open unit interval and receives NaN. Optics, antenna pattern analysis, and some control systems work with secant ratios or with reciprocals of direction cosines and then need the unsigned principal angle for constraints or orientation.
Surveying and construction workflows occasionally convert measured secant ratios into angles when only a reciprocal cosine is available from instrumentation. If you later need arccos alone, arcsine, arctangent, arccosecant, multi-branch inverses, or complex arcsecant, treat those as separate search intents rather than overloading this arcsecant-in-degrees landing.
What you can do with it
Homework and exam checks
Recover the principal arcsecant of a secant ratio in degrees, and reject values inside (−1, 1) with a clear error instead of NaN.
Engineering and orientation angles
Convert a secant or reciprocal direction-cosine ratio into the principal angle for constraints, sensors, and orientation pipelines.
FAQ
What does this calculator compute?
It returns the principal inverse secant of a real value x with |x| ≥ 1: the unique angle theta in [0, π] excluding π/2 such that sec(theta) = x, reported primarily in degrees (and also in radians).
What is the allowed input range?
The secant value must be a finite number with absolute value at least one.
How is arcsec related to arccos?
The service uses the identity arcsec(x) = acos(1/x) for |x| ≥ 1, then converts radians to degrees by multiplying by 180/π.
What is the principal range of arcsecant in degrees?
From 0° to 180° excluding 90°. That matches the standard single-valued branch obtained from acos(1/x).
What does it cost?
The browser tool is free and runs locally.
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/arcsec-degrees \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"value":2}'const res = await fetch("https://api.kit.forhosting.com/trig/arcsec-degrees", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"value": 2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/trig/arcsec-degrees",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"value": 2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/trig/arcsec-degrees", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"value":2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"value":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/trig/arcsec-degrees", 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": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "trig.arcsec_degrees",
"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
Values inside the open interval from negative one to one, non-finite numbers, and missing fields are rejected with invalid_input instead of a silent NaN. The free browser tool and the paid API share one pure function, so demos and automated scripts never diverge on the same input.
How inverse secant recovers the principal angle from a secant
The inverse secant undoes that map for a single principal value: for a real number x with absolute value at least one, arcsec of x is the unique angle theta in the closed interval from zero to pi excluding pi over two such that secant of theta equals x.
What you send and what you get back
The number must satisfy absolute value greater than or equal to one. Example checks: value two yields sixty degrees; value negative two yields one hundred twenty degrees; value one yields zero degrees; value negative one yields one hundred eighty degrees; value zero point five fails the domain gate. Wire the free browser tool for quick sanity checks, then call the identical contract from scripts when you need bulk arcsecant conversion without reimplementing the domain gate in every language. Provide one required finite number under the field value, with aliases x, sec, secant, and input for friendlier form labels. Missing fields, empty strings, non-numeric text, NaN, and Infinity are rejected with invalid_input and are not charged. If the number is finite but lies strictly inside the open interval from negative one to one, the error message names the field, the out-of-domain value, and the domain rule so clients can correct the input without guessing whether the endpoints plus or minus one were allowed. On success the JSON response echoes the normalized value, returns degrees as the principal arcsecant scaled from radians, returns radians as the intermediate acos of one over x in the range zero to pi excluding pi over two, sets unit_degrees to deg and unit_radians to rad, includes method as the string arcsecant, and includes formula as the identity arcsec of x equals acos of one over x for absolute value at least one.
Where inverse secant shows up in real work
Because this capability is a pure parse engine, API pricing is the published base of $0.002 per successful item and failed validation does not consume balance.
Integration use cases
Deterministic test fixtures
Generate stable arcsec(x) fixtures for unit tests and documentation with the same pure function in browser and API.