Sine half-angle calculator
Find sin(θ/2) with the right sign every time: 90° halves to 45° and gives a positive result, while −90° halves to −45° and gives a negative one. The quadrant of the half angle is shown too. 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
The sine half-angle identity is one of the first radical formulas students meet after the double-angle relations, and it shows up again in integration tables, Fourier work, and any geometry step that needs the sine of an unknown half of a measured angle. A raw square root only gives the magnitude; the sign of sin(θ/2) depends on which quadrant the half-angle lands in, so a calculator that always returns a positive radical will fail half of the homework checks and every signed sensor path. This sine half-angle calculator accepts a full angle θ in degrees or radians, evaluates cos θ, applies sin(θ/2) equals plus or minus the square root of (1 minus cos θ) over 2, and chooses the sign from the normalized half-angle quadrant so the numeric value matches the true sine of θ/2.
What the half-angle sine identity is and why the sign matters
The correct choice is not arbitrary: it is the sign of the actual sine of θ/2 on the unit circle. If θ is 90 degrees, then θ/2 is 45 degrees in the first quadrant and sin(θ/2) is positive one over square root of two. If θ is negative 90 degrees, then θ/2 is negative 45 degrees in the fourth quadrant and sin(θ/2) is negative.
Returning only the radical would give the same positive number for both inputs and break any check that compares against Math.sin of half the angle. The response also echoes the half-angle in both radians and degrees and labels the half-angle quadrant (or axis) so you can show students why the sign was chosen without re-deriving the unit-circle picture by hand.
What you can do with it
Classroom and homework checks
Let students submit θ in degrees or radians and verify the signed sin(θ/2) against the half-angle identity, including negative results in Q3 and Q4.
Exam and LMS auto-grading
Mark half-angle sine items with a deterministic oracle that never drops the quadrant sign or invents a positive radical for a negative half-angle.
Notebooks and signal pipelines
Evaluate sin(θ/2) from measured full angles in calibration scripts without importing a full CAS or re-implementing radical edge cases.
FAQ
How do I compute sin of half an angle?
Send a finite angle and optional unit (deg default, or rad).
Why is there a plus-or-minus in the half-angle formula?
The square root only gives the magnitude. The correct sign is the sign of the actual sin(θ/2) on the unit circle, which this calculator selects automatically.
Can I enter the angle in radians?
Yes. Set unit to rad or radians. Degrees use deg or degrees (default).
What is the sine of half of 90 degrees?
Half of 90° is 45°, which lies in the first quadrant, so the sine is positive: one over the square root of two. The calculator also shows the half angle in degrees and radians.
Why is the result negative for −90 degrees?
Because half of −90° is −45°, which falls in the fourth quadrant, where sine is negative. A bare square root would wrongly give the same positive number as for 90°.
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/sin-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/sin-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/sin-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/sin-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/sin-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.sin_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
What the half-angle sine identity is and why the sign matters
From the cosine double-angle formula cos θ equals 1 minus 2 sin squared of (θ/2), you rearrange to isolate the sine of the half-angle: the absolute value of sin(θ/2) equals the square root of (1 minus cos θ) divided by 2. That radical is always non-negative, so textbooks write the full identity with a plus-or-minus in front. This endpoint therefore computes the magnitude from cos θ and multiplies by plus one or minus one according to the sign of sin of the normalized half-angle, so the result is the true sin(θ/2) including zeros on the axes and full range values from negative one through positive one.
Input contract, units, response shape, and validation errors
Degrees are converted to radians with the standard factor π over 180 before any cosine is taken. POST a JSON body with a required finite number angle (aliases theta or value) and an optional unit string that defaults to deg. Accepted unit values are deg, degrees, rad, and radians; other strings fail with invalid_input. On success the body includes the echoed angle and unit, angle_rad and angle_deg for display in either system, half_angle_rad and half_angle_deg for the exact half, sin_half as the signed numeric sine rounded to twelve decimal places for stable goldens across JavaScript engines, cos_theta used inside the identity, half_quadrant as 1 through 4 or an axis label such as positive_x, and a short formula string restating the identity. Missing angle, empty strings, Infinity, NaN, and non-numeric types all fail with invalid_input naming the field; those failures are not billed. Example: angle 90 with unit deg yields half_angle_deg 45 and sin_half about 0.707106781187 in quadrant 1. Example: angle π with unit rad yields half_angle_rad about 1.5708 and sin_half equal to 1 on the positive y-axis. Pair the response with your own graphing if you need a sketch; this endpoint returns numbers only and does not emit plots or multi-step prose beyond the formula field.
Where a dedicated sine half-angle API earns its keep
Classrooms and tutoring platforms use a sine half-angle calculator so learners can check the signed radical against textbook answers without flipping the plus-or-minus by hand. Exam generators need a deterministic oracle that marks wrong a always-positive square root when the half-angle sits in the third or fourth quadrant. Failed validation does not consume balance. Physics and signal notebooks often reduce double-angle measurements to sin of half an angle when integrating, simplifying radicals, or matching phase-sensitive sensor readings; calling a tiny parse API avoids pulling a full computer-algebra system for one identity. Keep requests small, pin the rounded sin_half in tests, and you get reproducible signed half-angle sines for homework, LMS plugins, and edge workers without standing up symbolic math infrastructure of your own.
Technical questions
What happens if angle is missing or not a number?
The task fails with invalid_input and is not billed. Provide a finite number for angle (aliases theta, value).
What does the sine half-angle calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.