Arcsine calculator
The arcsine of 0.5 is 30°, or 0.5235987756 radians, always taken from the principal range between −90° and 90°. Both units come back together, and values outside −1 to 1 are refused rather than quietly clipped. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Value
- 0.5
- Radians
- 0.5235987756
- Degrees
- 30
- Range
- [-π/2, π/2] radians (−90° to 90°)
The arcsine calculator answers a single, everyday trigonometry question with one deterministic contract: given a real sine value between minus one and one, what is the principal angle whose sine equals that value? This tool returns that angle in both radians and degrees so you never have to remember whether your worksheet, CAD script, or physics library expects π units or a full three-hundred-sixty-degree circle. Classic checkpoints include zero mapping to zero, one half mapping to thirty degrees or π over six, the square root of two over two mapping to forty-five degrees, the square root of three over two mapping to sixty degrees, and plus or minus one mapping to plus or minus ninety degrees. The engine uses the principal branch from negative π over two to positive π over two, never samples randomness, never reads the clock, and never calls the network.
What the principal arcsine is and why the domain is closed
The arcsine function, written arcsin(x) or asin(x), is the inverse of the sine function on a restricted interval chosen so that the inverse is single-valued. On the real numbers the sine of an angle is never smaller than minus one and never larger than one, so any real input to arcsine must lie in that same closed interval.
Inside the interval the principal value is the unique angle θ between negative ninety degrees and positive ninety degrees — equivalently between negative π over two and positive π over two radians — such that sin(θ) equals x. That principal branch is what programming languages return from Math.asin, what scientific calculators label as sin⁻¹ in degree or radian mode, and what textbooks use when they ask for the inverse sine of a sine ratio without specifying a second quadrant solution.
This calculator follows that convention exactly: zero yields zero, positive inputs yield angles in the open first quadrant up to ninety degrees at x equals one, and negative inputs yield the matching negative angles down to minus ninety degrees at x equals minus one. Successful responses always include the echoed value, radians, degrees, a short range string, and the formula so callers can audit the result without opening a second reference page.
Where arcsine checks fit in study, engineering, and automation
Students meet arcsine when right-triangle ratios turn into missing angles, when unit-circle exercises ask for the angle with a given sine, and when physics problems recover an elevation angle from a vertical component. Teachers can generate answer keys that list both degree and radian forms without trusting opaque calculator apps that hide mode settings.
Surveyors, game developers, and robotics hobbyists often need to invert a normalized sine from a projection or a control signal; converting that ratio into a principal angle without a hand calculator or a forgotten degree-mode toggle avoids layout and control mistakes. Developers call the same contract from scripts to seed deterministic test vectors for math libraries, to illustrate inverse trigonometric functions in teaching tools, or to satisfy programming-challenge statements that demand exact asin completions within a tolerance.
Data pipelines that normalize orientation readings sometimes need to reject out-of-range sine samples before a downstream step assumes a real angle. Pair it with triangle solvers when you need full SSA or right-triangle completions, with slope tools when you need inclination from rise and run, and with other analytic helpers when your workflow mixes pure math with document or data steps.
What you can do with it
Homework and exam inverse-sine checks
Convert sine ratios into principal angles in degrees and radians for worksheets, answer keys, and contest practice without guessing calculator mode.
Teaching unit circle and right triangles
Show students how x maps to θ on the principal branch while they change inputs live in the browser at no cost.
FAQ
What does the arcsine calculator cost?
It is free in your browser on this page.
What happens if value is outside [-1, 1]?
A real arcsine is defined only on the closed interval from minus one to one, so out-of-range values are rejected rather than clamped.
Which angle branch is returned?
The principal branch: θ in [-π/2, π/2] radians, or equivalently −90° to 90°. Secondary solutions such as π − θ are not returned.
Are results available in both degrees and radians?
Yes. Every successful response includes radians from Math.asin and degrees converted with θ · 180 / π, rounded to the requested precision.
Is the algorithm randomized?
No. It uses deterministic floating-point arcsine and a fixed conversion factor. The same value and precision always yield the same radians and degrees.
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/inverse-sin \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"value":0.5}'const res = await fetch("https://api.kit.forhosting.com/trig/inverse-sin", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"value": 0.5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/trig/inverse-sin",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"value": 0.5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/trig/inverse-sin", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"value":0.5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"value":0.5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/trig/inverse-sin", 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": 0.5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "trig.inverse_sin",
"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
Inputs outside the closed interval from minus one to one are rejected with invalid_input because a real arcsine does not exist there. Explore free in the browser for homework and teaching demos, or call the API at $0.002 per successful request when grading scripts and pipelines need reproducible asin results.
What the principal arcsine is and why the domain is closed
Values strictly outside minus one to one fail with invalid_input and a message that names the domain; they are not clamped or wrapped, because silent clamping would hide bad sensor data and bad homework answers.
How inputs are read and how degrees and radians are produced
Decimal strings such as "0.5" or "-1" are accepted after trim; embedded spaces, hex prefixes, and malformed tokens are rejected so a bad cell does not become a wrong angle. Internally the engine evaluates Math.asin on the finite real, multiplies the radian result by one hundred eighty over π to obtain degrees, rounds both angles and the echoed value to the requested precision, and attaches range equal to the principal interval description plus formula θ equals arcsin(x) with the degrees conversion written out. Boundary points are first-class: arcsin of one is exactly π over two radians and ninety degrees; arcsin of minus one is the negatives of those; arcsin of zero is zero in both units. The required field is value as a JSON number, with aliases x, number, sin, and sine so forms, query strings, and spreadsheet exports can reuse one contract without renaming columns. Optional precision, with aliases decimals and dp, is an integer from zero through fifteen that defaults to ten decimal places on the numeric fields in the response. Common exact school values round cleanly at default precision: one half to thirty degrees, square root of two over two near forty-five, square root of three over two near sixty.
Where arcsine checks fit in study, engineering, and automation
On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split — free interactive exploration and a priced, identical API — matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result.
Integration use cases
Deterministic library and CI fixtures
Generate reproducible asin checkpoints for unit tests, property checks, and CI assertions without embedding static angle tables.