Trig identities calculator
At 30°, sin² + cos² comes out at exactly 1 and the double-angle formulas return the sine and cosine of 60°. Enter one angle in degrees or radians and see both identity families evaluated, with a check that each one holds. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Angle
- 30
- Radians
- 0.5235987756 deg
- Sin
- 0.5
- Cos
- 0.8660254038
- Sin2 plus cos2
- 1
Fundamental trigonometric identities are the algebraic relations that sine, cosine, and tangent always satisfy once the angle is inside each function's domain, and they are the first toolkit students reach for after the unit circle. The two families most homework sets and contest drills actually use are the Pythagorean identities, which start from sine squared plus cosine squared equals one, and the double-angle identities, which rewrite sine, cosine, and tangent of twice an angle in terms of the original angle. This calculator takes one finite measure in degrees or radians, evaluates both families at that angle, and returns the computed identity values together with residual checks so a learner or a grader can see that the identities hold numerically.
What Pythagorean and double-angle identities are, and what this page returns
In trigonometry, an identity is an equality that holds for every angle in the common domain of the functions it names. The Pythagorean family begins with sine squared of theta plus cosine squared of theta equals one, which is the unit-circle statement that a point whose coordinates are cosine of theta and sine of theta sits on the circle of radius one.
Dividing that relation by cosine squared or by sine squared produces the two companion forms one plus tangent squared equals secant squared, and one plus cotangent squared equals cosecant squared, each defined only where the divisor is nonzero.
The double-angle family is the next block in a typical syllabus: sine of two theta equals two times sine times cosine; cosine of two theta equals cosine squared minus sine squared, or the two equivalent rewrites two cosine squared minus one and one minus two sine squared; and tangent of two theta equals two tangent over one minus tangent squared whenever that denominator is nonzero. Search intent clusters around trig identities, trigonometric identities calculator, Pythagorean identity, sin two theta, and verify double angle formula.
Learners want the numeric values of those identities at a concrete angle they can check against a homework key, not only a chalkboard rewrite. This landing evaluates both families together so a single request returns the Pythagorean sum, the double-angle triple, and residual flags, which is a different job from a residual-only verifier or a double-angle-only triple.
What you can do with it
Homework identity drills
Students and graders confirm sin²(θ)+cos²(θ)=1 and the double-angle values at a degree or radian measure, with undefined companions omitted instead of returned as infinity.
Notebooks and animation headings
Scripts that need both a unit-circle check and sin(2θ), cos(2θ) from one heading call a pure identity table instead of reimplementing two formula families.
FAQ
Which trigonometric identities does this calculator evaluate?
The Pythagorean identity sin²(θ)+cos²(θ)=1, the companions 1+tan²(θ)=sec²(θ) and 1+cot²(θ)=csc²(θ) when defined, and the double-angle identities sin(2θ)=2sin(θ)cos(θ), cos(2θ)=cos²(θ)−sin²(θ), and tan(2θ) when defined. Send a finite angle.
What happens if the angle is not a finite number?
Missing, empty, non-numeric, infinite, and NaN angles are rejected and are not billed.
Does the angle have to be in degrees?
Degrees are the default. Set unit to rad when the measure is already in radians. Identity values are unitless; unit only scales how θ is interpreted.
What happens at 90° or 45°, where some identities are undefined?
Defined values are still returned. Tangent and the tan-sec companion are omitted at 90°; tangent of two theta is omitted at 45°, where cos(2θ) is zero. Those omissions are not billed failures.
What does the residual in the result mean?
It is the gap between sin² + cos² and 1 at your angle. A residual of zero, as at 30°, means the identity holds exactly; anything else would show by how much the numbers drift from the textbook rule.
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/math/trig-identities \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"angle":30}'const res = await fetch("https://api.kit.forhosting.com/math/trig-identities", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"angle": 30
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/trig-identities",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"angle": 30
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/trig-identities", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"angle":30}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"angle":30}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/trig-identities", 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": 30
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.trig_identities",
"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
A missing or non-finite angle fails with invalid_input and is not billed.
What Pythagorean and double-angle identities are, and what this page returns
Companion values such as tangent, cotangent, and tangent of two theta are omitted when they are undefined rather than returned as infinity, because a non-finite JSON number is a worse contract than a missing optional field.
How this calculator validates input, applies the identities, and shapes the response
Unit defaults to deg; set unit to rad when the measure is already in radians. Internally the angle is converted to radians only for the library sine and cosine, then the identities are applied exactly as written in a textbook: the Pythagorean sum is sine squared plus cosine squared, compared against one to produce a residual and a holds flag; sine of two theta is twice the product of sine and cosine; cosine of two theta is cosine squared minus sine squared; and those two identity values are compared against the library sine and cosine of the doubled measure so a client can assert that the rewrite matches the doubled-angle evaluation. Tangent of the original angle and the companion one plus tangent squared versus secant squared appear only when cosine is safely away from zero; cotangent and the companion one plus cotangent squared versus cosecant squared appear only when sine is safely away from zero; tangent of two theta appears only when cosine of two theta is safely away from zero. A typical success path is angle thirty degrees, which yields sine one half, cosine the square root of three over two, Pythagorean sum one, sine of sixty equal to the square root of three over two, and cosine of sixty equal to one half. POST a JSON body with required field angle and optional field unit. Angle must be a finite number; JSON numbers and numeric strings are accepted, and common aliases theta and x are read as the same field. Words, empty strings, null, infinity, and NaN fail with invalid_input that names the angle, and those failed calls are not charged. Results are rounded to a stable twelve-decimal precision so the same input always yields the same JSON, including for radian measures that cannot be represented exactly in binary floating point. The response also echoes angle, unit, and radians, reports double_angle as twice the input in the same unit, and includes short identity strings so a notebook can display the rewrite next to the numeric values.
Where a combined trig-identities API earns its keep, and what it costs
Use the free browser tool when you want an immediate identity table while checking homework, verifying a unit-circle sketch, or confirming that a double-angle rewrite still matches sine and cosine of twice the heading, without installing a computer-algebra system. There is no language model, no external algebra service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials. Each successful request is one billable item at $0.002; failed validation, including a missing or non-finite angle, does not consume balance. Accepted input is any finite real measure in degrees or radians; the response returns the computed Pythagorean and double-angle values, residual flags, and the optional companions only when they are defined. Results are fully deterministic: the same angle and unit always yield the same JSON. Limits are intentional: this endpoint does not expand a named squared function into a first-power cosine of the double angle, does not solve inverse-trig equations, and does not return exact symbolic radicals. Those are different search intents and live on their own landings. Wire the endpoint into identity drills, unit-circle demos, contest answer checkers, and animation helpers that need both a Pythagorean check and a double-angle triple from one heading.
Integration use cases
Lesson platforms and contest graders
A deterministic oracle marks Pythagorean and double-angle items from one JSON contract and rejects a non-finite angle with invalid_input rather than a silent empty payload.
Technical questions
What does the trig identities calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; a non-finite or missing angle is not billed.