Half-angle calculator
Enter a full angle of 90° and get a half-angle of 45° with sine and cosine of 0.707106781187 and a tangent of 1. Each sign follows the quadrant where the half-angle really lands, not just the positive square root. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Angle
- 90 deg
- Angle rad
- 1.5707963268 deg
- Angle deg
- 90
- Half angle rad
- 0.7853981634
- Half angle deg
- 45
Half-angle formulas appear whenever you need the sine, cosine, or tangent of θ/2 without calling a full trigonometric library on the halved value alone—for example when verifying textbook exercises, reducing multi-angle expressions, or checking that a firmware atan2 path agrees with closed-form identities. This capacity takes a full angle θ in degrees or radians, applies the classic identities sin(θ/2) = ±√((1 − cos θ)/2), cos(θ/2) = ±√((1 + cos θ)/2), and tan(θ/2) = sin θ/(1 + cos θ), and chooses each sign from the quadrant of the true half-angle so results match the geometric half-turn rather than an unsigned radical.
What the half-angle identities compute
Given a known full angle θ, the half-angle identities express the trigonometric functions of θ/2 in terms of cos θ and sin θ. The sine and cosine forms use square roots of (1 − cos θ)/2 and (1 + cos θ)/2 respectively, which always yield non-negative magnitudes; the correct algebraic sign is not free to invent—it must match the quadrant where the half-angle actually lands.
For example, if θ = 270° then θ/2 = 135°, which sits in the second quadrant, so sin(θ/2) is positive while cos(θ/2) is negative, and tan(θ/2) is therefore negative. A calculator that only returns the positive square root will look right on acute first-quadrant homework and wrong as soon as the half-angle crosses into another sector of the plane.
The tangent half-angle formula tan(θ/2) = sin θ/(1 + cos θ) (or the companion form (1 − cos θ)/sin θ when the first denominator vanishes) carries the sign automatically in almost all cases, and this tool falls back carefully when θ is an odd multiple of π where tan(θ/2) is undefined.
Using the identities rather than merely calling sin(θ/2) in isolation is useful pedagogically: students and reviewers can see that the result agrees with the radical form they wrote by hand, while still getting the engineering-correct signed values they need for vectors, phasors, and geometric constructions.
Signs, undefined tangent, precision, and limits
Sign selection is driven by the normalized half-angle in (−π, π]: sine is non-negative in the upper half-plane of that interval and cosine is non-negative in the right half-plane, matching standard unit-circle conventions. Magnitudes still come from the half-angle radicals so floating-point paths stay consistent with textbook identities even when intermediate cos θ is slightly outside [−1, 1] due to rounding—values under the square root are clamped at zero.
When cos θ is nearly −1 and sin θ is nearly 0, tan(θ/2) is reported as undefined rather than a huge accidental number.
What you can do with it
Homework and textbook half-angle checks
Students enter 60°, 90°, or 270° and confirm signed sin(θ/2), cos(θ/2), and tan(θ/2) against hand work that uses the radical identities.
Engineering reductions and firmware audits
Scripts verify that half-angle reductions in control law or CAD code match closed-form identities before shipping a numeric change.
FAQ
Which half-angle formulas do you use?
sin(θ/2) = ±√((1 − cos θ)/2), cos(θ/2) = ±√((1 + cos θ)/2), and tan(θ/2) = sin θ/(1 + cos θ) (with the companion form when needed). Signs follow the quadrant of θ/2.
What happens if the angle is not a number?
Only finite numeric angles (or digit strings that parse to finite numbers) are accepted.
When is tan of the half-angle undefined?
When θ/2 is an odd multiple of π/2 (for example θ = π), cosine of the half-angle is zero.
What are the half-angle values for 90 degrees?
The half-angle is 45°, so the sine and the cosine are both 0.707106781187 and the tangent is 1. The half-angle lies in the first quadrant, so all three values are positive.
How does the calculator choose the plus or minus sign?
From the quadrant where the half-angle actually lands. For 270°, the half-angle is 135° in the second quadrant, so the sine is positive, the cosine negative and the tangent negative.
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/half-angle \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"angle":90}'const res = await fetch("https://api.kit.forhosting.com/trig/half-angle", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"angle": 90
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/trig/half-angle",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"angle": 90
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/trig/half-angle", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"angle":90}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"angle":90}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/trig/half-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": 90
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "trig.half_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
Non-numeric input is rejected with invalid_input so bad data never looks like a real answer.
How to call the calculator and read the result
Send a numeric field named angle (aliases theta and value are accepted) and optionally unit set to deg or degrees (the default) or rad or radians. Digit strings that parse to finite numbers are allowed; missing fields, empty strings, NaN, Infinity, objects, and other non-numeric types fail with invalid_input naming the problem. On success the JSON echoes the input angle and unit, reports the full angle in both radians and degrees, reports the half-angle in both units, and returns sin_half, cos_half, and tan_half together with sin_theta and cos_theta for audit. When the half-angle lands on an odd multiple of π/2, tan_half is null and tan_defined is false so clients do not treat an infinite slope as zero. A half_quadrant label (1–4 or axis tags such as positive_y) makes it easy to spot-check the sign choice without redrawing the unit circle. Pricing for a successful API call is a flat $0.002 under the item unit; validation failures are never charged. Typical clients include students checking homework for 60°, 90°, or 270° cases, tutors generating answer keys, engineers verifying half-angle reductions in CAD or control software, and automated tests that freeze expected JSON for regression.
Signs, undefined tangent, precision, and limits
Related search intents such as double-angle formulas, full sin/cos/tan of an angle without halving, or polar conversions are separate landings; this page is dedicated to half-angle sin, cos, and tan with correct signs from a single full angle.
Integration use cases
CI fixtures for trig libraries
Pipelines freeze JSON golden outputs for positive and negative half-angles so a future refactor cannot flip signs silently.
Technical questions
How do I calculate sin, cos, and tan of half an angle?
POST a numeric angle (and optional unit deg or rad) to the half-angle endpoint, or use the free browser form. The response includes sin_half, cos_half, and tan_half with quadrant-correct signs.
Is the browser half-angle calculator free?
Yes. The in-browser runner uses the same library as the API and costs nothing. The API charges $0.002 per successful request from your prepaid KIT balance.