Coterminal angle calculator
Type 45° and you get 45° as the positive coterminal angle and −315° as the negative one, both within a single turn of the original ray. Degrees and radians are both accepted, and the same angle always returns the same pair. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Given angle
- 45
- Revolution
- 360 deg
- Positive
- 45
- Negative
- -315
Two angles are coterminal when they share the same terminal side after both are drawn in standard position on the coordinate plane. Algebraically that means they differ by an integer number of full turns: three hundred sixty degrees, or two pi radians. Homework and contest problems almost always ask for one positive coterminal and one negative coterminal that each stay inside a single revolution of the original ray, not for the infinite family of answers you would get by adding every multiple of a full turn. Students checking unit-circle worksheets, teachers building answer keys, and scripts that normalize heading angles all need a deterministic pair they can pin in continuous integration.
What coterminal angles are and why one-revolution answers matter
In trigonometry, an angle in standard position has its vertex at the origin and its initial side along the positive x-axis. The terminal side is the ray you reach after rotating by the given measure. Any other measure that lands on that same ray is coterminal with the first: you may wind forward or backward any whole number of complete revolutions without changing the terminal direction.
Search intent clusters around phrases such as coterminal angle calculator, positive and negative coterminal angles, angles within one revolution, and find coterminal of negative degrees. Learners want a concrete pair they can write beside a figure—typically the unique positive representative in the half-open interval from zero up to but not including three hundred sixty degrees, and the unique negative representative in the interval from minus three hundred sixty up through zero—rather than an abstract congruence class.
The same idea appears when pilots normalize headings, when game engines wrap yaw, and when unit-circle drills ask for an equivalent acute or obtuse measure without drawing another sketch.
How this calculator validates input, reduces the angle, and reports both sides
Unit defaults to deg for a full turn of three hundred sixty; set unit to rad for a full turn of two pi. The engine computes the positive remainder of the angle modulo one revolution, mapping every real measure into the half-open interval from zero inclusive to the revolution size exclusive.
The negative coterminal is that positive value minus one full turn, except when the positive reduction is exactly zero: both positive and negative are reported as zero because the terminal ray is the positive x-axis itself, which is the same side as plus or minus a full turn.
Edge cases are explicit: four hundred five degrees reduces to forty-five positive and minus three hundred fifteen negative; minus forty-five becomes three hundred fifteen positive and minus forty-five negative; seven hundred twenty degrees and minus two pi both land on the zero pair.
What you can do with it
Homework coterminal drills within one revolution
Students and graders confirm the positive and negative coterminal pair for degree or radian measures, including clear rejection when the angle field is non-numeric.
FAQ
What does within one revolution mean here?
Positive is the unique coterminal in [0, full) and negative is the unique coterminal in (−full, 0], where full is 360° or 2π depending on unit. Multiples of a full turn map both to zero.
What happens if the angle is not a number?
Non-numeric or non-finite values are rejected before any reduction runs; those calls are not charged.
Is the coterminal angle calculator free?
The browser tool runs free on your device.
Can I use radians instead of degrees?
Yes. Set unit to rad so one revolution is 2π. Omit unit or set deg for a 360° full turn.
How do I get the negative coterminal angle?
Take the positive coterminal angle and subtract one full turn. For 45°, a full turn is 360°, so the negative partner is −315°. If the positive value is exactly zero, both answers are reported as zero.
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/coterminal-angle \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"angle":45}'const res = await fetch("https://api.kit.forhosting.com/trig/coterminal-angle", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"angle": 45
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/trig/coterminal-angle",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"angle": 45
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/trig/coterminal-angle", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"angle":45}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"angle":45}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/trig/coterminal-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
{
"angle": 45
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "trig.coterminal_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
This capacity accepts a finite angle in degrees or radians, reduces it with modular arithmetic into the half-open positive interval from zero inclusive to one full turn exclusive, derives the matching negative partner in the open-closed interval from minus one full turn to zero inclusive, and returns both values as structured JSON. Non-numeric or non-finite input is rejected with invalid_input and is not charged.
What coterminal angles are and why one-revolution answers matter
Delivering both representatives as JSON makes the result easy to paste into notes, diff in automated tests, or feed into a larger pipeline without reimplementing modular reduction and sign handling in every language you teach or deploy.
How this calculator validates input, reduces the angle, and reports both sides
Send a JSON body with required field angle and optional field unit. Angle must be a finite number—JSON numbers and numeric strings such as forty-five or negative one hundred twenty are accepted; words, empty strings, null, infinity, and NaN fail with invalid_input so those failed calls are not charged. Results are rounded to a stable ten-decimal precision so the same input always yields the same JSON, including for irrational radian measures.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate positive and negative coterminal pair while checking homework, labeling a unit-circle sketch, or confirming a heading wrap without installing a computer-algebra system. Results are fully deterministic: the same angle and unit always yield the same positive and negative pair. There is no language model, no external geometry service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials. Each successful request is one billable item at $0.002; failed validation does not consume balance. Accepted input is any finite real measure in degrees or radians; the response echoes the parsed angle and unit, reports the revolution size used, and returns both coterminal representatives with method coterminal_one_revolution. Wire the endpoint into unit-circle demos, contest answer checkers, navigation preprocessing that prefers a signed short wrap, or floating pipelines where you already prefer explicit domain errors for non-numeric payloads over opaque black-box widgets that cannot show why an angle failed to parse.
Integration use cases
Deterministic fixtures for trigonometry tests
Pipelines that seed unit tests or interactive unit-circle demos can call a pure coterminal oracle instead of reimplementing modular wrap logic in every language.
Heading and yaw normalization helpers
Scripts that wrap compass headings or game yaw into one signed revolution recover both the non-negative and negative representatives as structured JSON for logs or UI.
Technical questions
How do I find coterminal angles with this API?
Send angle (required) and optional unit deg or rad to trig.coterminal_angle. The response returns positive and negative coterminal measures within one revolution.