Solve sine equation calculator
Find every angle where the sine takes your value, not just the first: a sine of 0.5 between −180° and 540° gives 30°, 150°, 390° and 510°. Both general families are shown with their 360° period. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Target
- 0.5
- Interval start
- -180
- Interval end
- 540
- Precision
- 6
- Equation
- sin(theta) = 0.5
- Angle unit
- deg
- Principal angle degrees
- 30
- Solutions
- 30, 150, 390, 510
This solve sine equation calculator finds every real angle that satisfies sin(theta) = target over an interval you choose. Enter a target from minus one through one, set inclusive starting and ending angles in degrees, and receive both periodic general solution families and the complete sorted list of specific answers inside that interval. Unlike a basic inverse sine button, it includes the second unit-circle branch and repeats both branches through as many full rotations as the interval contains. Invalid real targets are rejected clearly instead of producing a misleading angle.
Why inverse sine alone does not solve the whole equation
Pressing inverse sine gives the principal angle alpha, but that is only the starting point for solving a sine equation. Sine is periodic and has the same value at two reflected positions in most rotations. The calculator normalizes those starting positions into one standard turn from zero up to, but not including, 360 degrees.
That makes the general result easier to read without losing negative or multi-turn answers, because integer values of k still generate every real solution. Special target values need care. At target 1, both formulas describe the same 90-degree family; at target -1, they describe the same 270-degree family.
The engine deduplicates those coincident branches instead of presenting one mathematical family twice. At target zero the two distinct bases are zero and 180 degrees, which correctly captures every integer multiple of 180. This distinction between a principal inverse and a complete periodic solution is the main reason to use an equation solver rather than an ordinary arcsine calculator.
How to choose an interval and read every returned field
Provide target as a finite real number in the closed interval from minus one to one. The interval may span multiple rotations, up to 36,000 degrees, so the solver remains bounded while still supporting one hundred complete turns. Optional precision controls zero through twelve decimal places in returned numeric fields and defaults to ten.
The response identifies the equation and degree unit, echoes the validated target, reports the principal angle, lists each general solution as a base, a 360-degree period, and a readable expression with integer k, and echoes the inclusive interval. An empty list is a valid result when the equation has real solutions but none falls inside the requested interval. Small numerical tolerances protect exact boundaries such as 30, 90, 180, and 360 from floating-point conversion noise.
Together, these rules make the result suitable for a worksheet, an answer checker, or a deterministic software test without hidden calculator-mode assumptions.
Domain checking, practical uses, and reproducible automation
A real sine can never be less than minus one or greater than one. Missing values, malformed numeric strings, Infinity, NaN, reversed intervals, excessive interval widths, and precision values outside the documented integer range are rejected in the same explicit way. Valid boundary targets are accepted: plus one produces 90 degrees plus full rotations, and minus one produces 270 degrees plus full rotations after normalization.
These checks are useful in learning platforms, where students should see whether their equation is impossible over the reals; in engineering scripts, where a sensor ratio slightly outside its physical range should stop the pipeline; and in test suites, where a stable oracle should enumerate all expected roots rather than only the first one. The algorithm is pure trigonometric arithmetic with bounded integer enumeration. It does not call a language model, contact a remote service, read the clock, use randomness, or store submitted values.
Use this solver when the question asks for all theta values on a stated degree interval, and use a principal arcsine tool only when one inverse-function value is genuinely all you need.
What you can do with it
Complete trigonometry homework checks
Find both unit-circle branches and every repeated solution on the interval specified by a worksheet or exam problem.
Build deterministic answer keys
Generate sorted degree solutions and a count for grading tools without maintaining hand-written tables of periodic roots.
Validate bounded physical models
Reject impossible sine targets from normalized measurements and enumerate every candidate angle within an operating range.
FAQ
Why are there usually two general solutions?
Within each 360-degree turn, sine has the same target value at alpha and 180 degrees minus alpha. Adding 360k to both produces every real solution.
Can the interval contain negative angles or several rotations?
Yes. Negative boundaries and multi-turn intervals are supported, provided the end is not before the start and the width is at most 36,000 degrees.
What are all the solutions of sin θ equal to 0.5?
They are 30° and 150° plus any whole number of full turns of 360°. Between −180° and 540°, that gives four answers: 30°, 150°, 390° and 510°.
Why is there only one family when the sine is 1 or −1?
At those values the two families land on the same angles, 90° for a sine of 1 and 270° for −1. The calculator removes the duplicate instead of listing the same family twice.
What happens if I enter a value greater than 1?
No real angle has a sine above 1 or below −1, so a value such as 1.2 is refused with a clear message naming the allowed range, instead of being quietly clipped.
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/solve-sin-equation \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"target":0.5}'const res = await fetch("https://api.kit.forhosting.com/trig/solve-sin-equation", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"target": 0.5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/trig/solve-sin-equation",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"target": 0.5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/trig/solve-sin-equation", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"target":0.5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"target":0.5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/trig/solve-sin-equation", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"target": 0.5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "trig.solve_sin_equation",
"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.
Limits
max_interval_degrees | 36000 |
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
It also handles negative angles, boundary endpoints, and the single-family cases at sine equal to plus or minus one. The browser calculation is free, while a successful API request costs $0.002.
Why inverse sine alone does not solve the whole equation
If alpha is the principal arcsine in degrees, one family is theta = alpha + 360k and the other is theta = 180 - alpha + 360k, where k is any integer. For example, sin(theta) = 0.5 has bases at 30 degrees and 150 degrees, so an interval from zero through 720 includes 30, 150, 390, 510, and endpoint rules determine whether any boundary angle is retained.
How to choose an interval and read every returned field
Then set interval_start and interval_end in degrees; both boundaries are inclusive, the start may be negative, and the end must not precede the start. If you omit the boundaries, the calculator uses zero through 360 degrees, a common textbook interval that intentionally contains both zero and 360 when each is a valid endpoint. The solutions array contains every matching degree value in ascending order, while solution_count makes empty or large lists easy to inspect programmatically. Rounding is applied to output values after the integer-period enumeration, so it does not decide whether an angle belongs near an endpoint.
Domain checking, practical uses, and reproducible automation
Therefore a target such as 1.2 has no real-angle solution, and this capability returns invalid_input with a message naming the closed domain rather than clamping the target or switching silently to complex numbers. Each successful API request is billed as one item at $0.002; validation failures are not charged.
Technical questions
What does the sine equation calculator cost?
It runs free in your browser. A successful API request costs $0.002; invalid inputs are not charged.
Are interval endpoints included?
Yes. Both interval_start and interval_end are inclusive, so a solution exactly on either boundary is returned.
What happens when the target is outside [-1, 1]?
The request fails with invalid_input because no real angle can have a sine outside the closed interval from minus one to one.