Unit circle calculator
At 30° the unit circle point is (√3/2, 1/2), and the angle is π/6 in radians. Enter any finite angle to get its cosine and sine as a point on the circle of radius 1, with exact forms for the standard angles. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Angle
- 30
- Input angle
- 30
- Input unit
- degrees
- Degrees
- 30
- Radians
- 0.5235987756
The unit circle calculator answers a core trigonometry question with one deterministic contract: given a finite real angle, what is the point (cos θ, sin θ) on the circle of radius one, and what is the same angle in radians? Classic school checkpoints include zero degrees at (1, 0), thirty degrees at (√3/2, 1/2), forty-five at (√2/2, √2/2), sixty at (1/2, √3/2), ninety at (0, 1), and the matching points in every other quadrant through three hundred sixty degrees. By default the engine returns textbook exact forms for those standard angles and the corresponding radian labels such as π over six or three π over two. The engine never samples randomness, never reads the clock, and never calls the network.
What the unit circle encodes and why cosine and sine are coordinates
The unit circle is the set of points at distance one from the origin in the plane. If you measure an angle θ from the positive x-axis, counterclockwise for positive angles, the terminal ray meets the circle at a unique point whose Cartesian coordinates are defined to be (cos θ, sin θ). That geometric definition is why cosine is the adjacent-over-hypotenuse ratio on a right triangle in the first quadrant and why sine is the opposite-over-hypotenuse ratio: both ratios become the x and y of a point on radius one.
A full turn is three hundred sixty degrees or two π radians; a right angle is ninety degrees or π over two. Negative angles reverse direction, and coterminal angles that differ by full turns land on the same point, so normalizing into a single turn is enough to read the table of standard values.
This tool applies that definition without calculator mode confusion: degrees and radians are both reported, the numeric cos and sin are rounded to the requested precision, and for the standard school angles the exact radical forms appear alongside the decimals. Axes are first-class: zero and three hundred sixty share (1, 0), ninety is (0, 1), one hundred eighty is (−1, 0), and two hundred seventy is (0, −1).
Where unit-circle lookups fit in study, design, and automation
Students meet the unit circle when filling cos and sin tables, when converting between degree diagrams and radian formulas, and when checking that a reference angle plus quadrant signs recovers the same coordinates as a calculator in the correct mode. Teachers can generate answer keys that show both the exact radical pair and the decimal point without trusting opaque apps that hide degree versus radian mode.
Game developers and robotics hobbyists often need the direction vector (cos θ, sin θ) for a heading; feeding a shared contract reduces silent off-by-π bugs when human-facing degrees meet engine radians. Surveyors and architects still annotate drawings in degrees while analysis scripts prefer radians for arc length and angular velocity; reporting both units with one request keeps those worlds aligned. Data pipelines that normalize orientation readings can reject non-numeric cells, optionally require standard angles for exact export, or fall back to numeric mode for continuous sensor streams.
Pair it with degree–radian converters when you only need unit changes, with inverse trigonometric helpers when you need angles from ratios, and with triangle solvers when you need full side-angle completions beyond a single unit-circle point.
What you can do with it
Homework and exam unit-circle tables
Fill cos and sin for standard angles with textbook exact forms and matching radian labels without guessing calculator mode.
Teaching coordinates and quadrant signs
Show students how the same angle maps to (cos, sin), exact radicals, and radians while they change inputs live in the browser at no cost.
FAQ
What does the unit circle calculator cost?
It is free in your browser on this page.
Which angles have exact values?
Standard unit-circle angles in the 30° and 45° families (0°, 30°, 45°, 60°, 90°, and their coterminal and quadrant partners). Exact mode returns forms like √3/2 and π/6 for those angles.
What if I need coordinates for a non-standard angle?
Set exact to false. The response then includes numeric cos, sin, and radians for any finite angle.
Can I enter the angle in radians?
Yes. Set unit to radians (or rad). The response always includes both degrees and radians for either input unit.
Is the algorithm randomized?
No. It uses a fixed standard-angle table and deterministic cos/sin evaluation. The same angle, unit, exact flag, and precision always yield the same result.
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/unit-circle \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"angle":30}'const res = await fetch("https://api.kit.forhosting.com/trig/unit-circle", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"angle": 30
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/trig/unit-circle",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"angle": 30
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/trig/unit-circle", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"angle":30}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"angle":30}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/trig/unit-circle", 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": 30
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "trig.unit_circle",
"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
If you request exact values for a non-standard angle, the request fails with invalid_input instead of inventing a fake radical; set exact to false for a numeric approximation of any finite angle. Explore free in the browser for homework, teaching demos, and CAD checks, or call the API at $0.002 per successful request when scripts and pipelines need reproducible unit-circle coordinates.
What the unit circle encodes and why cosine and sine are coordinates
Quadrant signs follow the usual pattern — both positive in the first, cosine negative and sine positive in the second, both negative in the third, cosine positive and sine negative in the fourth — so students can check hand work against a single stable payload rather than flipping between apps that disagree on exact versus approximate display.
How inputs are read and when exact mode accepts or rejects an angle
Coterminal angles such as three hundred ninety degrees or negative thirty degrees match the same table rows as thirty and three hundred thirty. The required field is angle as a JSON number, with aliases value, degrees, deg, theta, and x so forms, query strings, and spreadsheet columns can reuse one contract. Optional unit, with aliases from and input_unit, accepts degrees, radians, deg, or rad and defaults to degrees so the primary classroom journey matches protractors and textbooks. Decimal strings such as "30" or "-45.0" are accepted after trim; hex prefixes, empty strings, and non-numeric tokens are rejected with invalid_input because the specification forbids inventing angles from garbage. Optional exact, with aliases exact_values and symbolic, defaults to true: the engine then returns cos_exact, sin_exact, radians_exact, and point_exact for standard angles in the thirty-degree and forty-five-degree families after normalizing the measure into the half-open interval from zero to three hundred sixty degrees. If exact is true and the normalized angle is not standard — for example seventeen degrees or one radian that is not a simple multiple of π over six or four — the request fails with invalid_input and a message that names the standard families and suggests exact equals false. When exact is false, Math.cos and Math.sin supply numeric coordinates for any finite angle while exact string fields stay null and is_standard reports whether a table match still exists. Optional precision, with aliases decimals and dp, is an integer from zero through fifteen that defaults to ten decimal places on the numeric fields.
Where unit-circle lookups fit in study, design, 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 unit-circle checkpoints for unit tests, property checks, and CI assertions without static hand tables.