Pythagorean identity sin cos calculator
Recover the missing ratio with the right sign: a sine of 0.6 in quadrant II gives a cosine of −0.8, because cosine is negative there. You also get an identity check that adds back to 1. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Value
- 0.6
- Known
- Sin
- Quadrant
- 2
- Precision
- 6
- Result
- -0.8
- Missing
- cos
- Sin
- 0.6
- Cos
- -0.8
This Pythagorean identity sin cos calculator finds cosine when sine is known, or sine when cosine is known. Enter the known function, its value, and the angle's quadrant. The calculator uses sin squared theta plus cos squared theta equals one to determine the missing magnitude, then applies the positive or negative sign required by the quadrant. It also returns both function values and a numerical identity check.
How the Pythagorean identity finds a missing trig value
The fundamental Pythagorean trigonometric identity says that sin squared theta plus cos squared theta equals one. If sine is known, rearrange the equation to cos squared theta equals one minus sin squared theta. Taking a square root gives the magnitude of cosine.
If cosine is known, rearrange it in the other direction: sin squared theta equals one minus cos squared theta, then take the square root to get the magnitude of sine. The important word is magnitude. A square root alone gives a nonnegative principal result, but an angle can have a negative sine or cosine.
That is why this calculator requires the quadrant as well as the known value. It computes the square root of one minus the known value squared, then chooses the appropriate sign from the quadrant. For example, a sine of 0.6 gives a cosine magnitude of 0.8.
In quadrant I the cosine is 0.8, while in quadrant II it is -0.8.
Use the quadrant to select the correct sign
Quadrants determine signs through the unit circle. In quadrant I, both sine and cosine are positive. In quadrant II, sine is positive and cosine is negative.
In quadrant III, both are negative. In quadrant IV, sine is negative and cosine is positive. The calculator validates the sign of the value you provide before deriving anything.
The same safeguard catches a negative sine in quadrant I. Values must be finite, nonzero, and strictly between -1 and 1. Requiring an interior-quadrant value keeps the input statement mathematically consistent and ensures the requested quadrant supplies an unambiguous sign.
Choose known as sin, sine, cos, or cosine; enter quadrant as an integer from 1 through 4; and optionally choose zero through fifteen decimal places with precision.
Check homework, lesson examples, and automated calculations
This calculator is useful whenever a problem gives one trigonometric ratio and enough location information to determine the other. A student solving a unit-circle exercise can enter the supplied sine, select the stated quadrant, and compare the derived cosine with a hand calculation. A teacher can demonstrate why plus-or-minus appears during algebra and why the quadrant resolves that choice.
Worksheet authors can generate consistent answer keys for decimal problems, while tutoring software can reject impossible sign-and-quadrant combinations before showing feedback. The computation uses no network, stored state, randomness, or current time. Running it interactively in the browser is free.
Remember that the output precision controls presentation, not the underlying mathematical method. If your source value is a rounded measurement, the derived value inherits that measurement uncertainty; displaying more decimal places cannot restore information that was absent from the original value. For exact radical exercises, use the decimal answer as a check rather than as a replacement for the requested symbolic derivation.
What you can do with it
Check a trigonometry homework answer
Enter the given sine or cosine and quadrant to verify the missing signed value after doing the identity algebra by hand.
Build consistent worksheet answer keys
Generate deterministic decimal answers and identity checks for exercises that provide one ratio and an interior quadrant.
Validate normalized coordinates in code
Reconstruct the missing unit-circle coordinate while rejecting values whose sign conflicts with the supplied quadrant.
FAQ
What formula does the calculator use?
It uses sin²θ + cos²θ = 1, rearranges for the missing squared function, takes the square root, and applies the sign required by the quadrant.
Why is the quadrant required?
The square root determines only a magnitude. The quadrant determines whether the missing sine or cosine is positive or negative.
Why are 0, 1, and -1 rejected?
Those values place the angle on an axis, and axes are not inside quadrants. This calculator accepts only values consistent with an interior quadrant.
What happens if the known value has the wrong sign?
For example, cosine cannot be positive in quadrant II, and sine cannot be negative in quadrant I.
How much does the calculator cost?
It is free to use in the browser.
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/pythagorean-identity-sincos \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"known":"sin","value":0.6,"quadrant":2}'const res = await fetch("https://api.kit.forhosting.com/trig/pythagorean-identity-sincos", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"known": "sin",
"value": 0.6,
"quadrant": 2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/trig/pythagorean-identity-sincos",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"known": "sin",
"value": 0.6,
"quadrant": 2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/trig/pythagorean-identity-sincos", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"known":"sin","value":0.6,"quadrant":2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"known":"sin","value":0.6,"quadrant":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/trig/pythagorean-identity-sincos", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"known": "sin",
"value": 0.6,
"quadrant": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "trig.pythagorean_identity_sincos",
"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
The calculation is deterministic, runs free in your browser, and costs $0.002 per successful API request when you need the same result in software, worksheets, or automated learning tools.
How the Pythagorean identity finds a missing trig value
The arithmetic under the square root is protected against tiny floating-point underflow, and the returned identity_check lets you confirm that the unrounded internal sine and cosine squares add back to one. This is a numerical calculator, so it returns decimals rather than symbolic radicals such as square root of three over two.
Use the quadrant to select the correct sign
A positive cosine paired with quadrant II, for instance, is contradictory and produces an invalid_input error instead of a plausible-looking answer. Although zero and the endpoints -1 and 1 are valid sine or cosine values at certain angles, those angles lie on the coordinate axes rather than inside a quadrant. The response normalizes the function name to sin or cos and identifies the missing function explicitly, so an API consumer does not need to infer what the result field represents.
Check homework, lesson examples, and automated calculations
In engineering or graphics code, the API can reconstruct one normalized coordinate from the other when the quadrant is already known, with a stable response shape that includes sin, cos, result, missing, and identity_check. A successful API call costs $0.002, while validation errors are returned as invalid_input.