General solution of sine equation
For sin θ equals 0.5, the principal angle is 30°, or 0.5235987756 radians, and every other solution follows from it by taking its supplement and adding whole turns of 360°. Both the nπ form and the two-family form are given, in degrees and radians. Free in your browser.
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Worked example
- Value
- 0.5
- Equation
- sin(θ)=0.5
- Principal radians
- 0.5235987756
- Principal degrees
- 30
- Formula npi radians
- θ = nπ + (−1)^n · α, n ∈ ℤ, α = arcsin(a)
The general solution of a sine equation answers the question every textbook and exam paper asks after you find one angle: if sin(θ) equals a given real number a, what is the full infinite family of real angles that work? This calculator takes a in the closed interval from minus one to one, computes the principal arcsine α, and returns the classic n-pi form θ equals n pi plus negative one to the n times alpha, together with the matching degree form and the equivalent two-family form in both units. Concrete sample angles for a small range of integers n let you check the first few solutions by hand or against a unit circle.
What sin(θ) = a means and the nπ plus-minus family
On the real line the sine function is periodic with period two pi and odd about the origin, so for any fixed a between minus one and one inclusive there are infinitely many angles whose sine is a. The principal solution is the unique alpha in the closed interval from negative pi over two to positive pi over two such that sin(alpha) equals a; that is exactly the arcsine of a returned by scientific calculators and by Math.asin.
Every other real solution is generated from that principal value by reflecting across the unit circle and adding full half-turns. The compact textbook form that encodes both reflections is theta equals n pi plus negative one to the power n times alpha, where n runs over all integers. When n is even the sign is positive and you recover alpha plus even multiples of pi; when n is odd the sign flips and you recover the supplementary angle in the second quadrant plus odd multiples of pi.
Equivalently you may write two separate families: theta equals two k pi plus alpha, or theta equals two k pi plus pi minus alpha, with k integer. Those two families together cover the same set as the single n-pi formula. In degrees the same identities hold after replacing pi with one hundred eighty degrees and two pi with three hundred sixty degrees.
Special checkpoints include a equals zero (solutions at integer multiples of pi), a equals one half (alpha equals thirty degrees or pi over six), a equals one (alpha equals ninety degrees or pi over two), and a equals minus one (alpha equals minus ninety degrees).
What you can do with it
Classroom general-solution checks
Let students enter a for sin(θ)=a and verify the nπ + (−1)^n α form plus sample angles in degrees and radians against textbook methods.
Exam and LMS auto-grading
Mark sine-equation general-solution items with a deterministic oracle that rejects |a|>1 and returns stable rounded sample angles.
Notebooks and phase recovery
Expand a measured sine ratio into the full real angle family in scripts without importing a full computer-algebra library.
FAQ
What is the general solution of sin(θ) = a?
For a in [-1, 1], let α = arcsin(a). Then θ = nπ + (−1)^n · α for every integer n. Equivalently θ = 2kπ + α or θ = 2kπ + (π − α) for integer k.
How do degrees and radians differ in the formulas?
The structure is the same: replace π with 180° and 2π with 360°. The principal α is converted with α° = α · 180/π. Sample angles are listed in both units for the same integers n.
How do I solve sin θ equals 0.5?
The principal angle is 30°. The full set is 30° plus any multiple of 360°, together with its supplement, 180° minus 30°, plus any multiple of 360°. Sample solutions for a few whole numbers are listed for checking.
What happens if the value is greater than 1?
There is no real solution, because the sine of a real angle always stays between −1 and 1. Such a value is refused with a clear message instead of an invented angle.
Why does the general solution have two families of angles?
Because two angles in each full turn share the same sine: the principal angle and its supplement. For a value of 0.5 those are 30° and 180° minus 30°, and adding whole turns of 360° to each gives every solution.
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/general-solution-sine \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"value":0.5}'const res = await fetch("https://api.kit.forhosting.com/trig/general-solution-sine", {
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/general-solution-sine",
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/general-solution-sine", 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/general-solution-sine", 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.general_solution_sine",
"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 with absolute value greater than one are rejected as invalid_input because sine never leaves that range on the reals.
What sin(θ) = a means and the nπ plus-minus family
This endpoint always returns both the symbolic n-pi form with the numeric alpha plugged in and the two-family restatement so you can copy the form your syllabus prefers.
Input contract, response fields, and error behaviour
POST a JSON body with the required finite number value equal to the right-hand side a of sin(theta) equals a. Aliases a, rhs, sine, and sin are accepted so spreadsheets and forms can reuse columns without renaming. The value must lie in the closed interval from minus one to one; outside that interval the sine equation has no real solution and the task fails with invalid_input naming the domain. Optional precision, with aliases decimals and dp, is an integer from zero through fifteen that defaults to ten and rounds every numeric field in the response. On success the body includes the echoed value, an equation string such as sin(θ)=0.5, principal_radians and principal_degrees for alpha, formula_npi_radians and formula_npi_degrees stating the abstract n-pi rules, formula_two_family_radians and formula_two_family_degrees for the two-branch form, general_solution_radians and general_solution_degrees with the numeric alpha substituted, samples_n plus samples_radians and samples_degrees for the requested integers, and a short formula summary string. Missing fields, non-numeric tokens, Infinity, and NaN all fail with invalid_input and are not billed. Pair the symbolic families with your own graphing if you need a sketch of y equals sin(theta) against the horizontal line y equals a; this endpoint solves for the angle family only and does not emit plots.
Where a general sine solution API earns its keep
Classrooms and tutoring platforms use a general solution of sine equation calculator so learners can check both the n-pi form and a short list of sample angles without losing a sign when they move from the principal arcsine to the full family. Failed validation does not consume balance. Keep requests small, pin the rounded samples in tests, and you get reproducible general solutions for homework, contest keys, and edge workers without standing up symbolic math infrastructure of your own. Physics and engineering notebooks often need every angle in a period that satisfies a measured sine ratio—for example elevation from a vertical component, phase from a normalized signal, or mechanical linkage angles—and calling a tiny parse API avoids pulling a full computer-algebra system for a one-parameter problem.
Technical questions
What happens if |a| is greater than 1?
The task fails with invalid_input. Sine never takes values outside [-1, 1] on the reals, so sin(θ)=a has no real solution and the call is not billed.
What do the sample_n and precision fields do?
sample_n (default 2) lists θ for n from −sample_n to +sample_n. precision (default 10, max 15) sets decimal places on all numeric fields for stable goldens and display.
What does the general solution of sine equation calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.