Six trigonometric functions from radians
At π/6 the six values are sine 1/2, cosine √3/2, tangent 1/√3, cosecant 2, secant 2/√3 and cotangent √3. Enter one angle in radians and get all six trigonometric functions in a single table. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Angle
- 0.5235987756
- Angle
- 0.5235987756
- Sine
- 0.5
- Cosine
- 0.8660254038
- Tangent
- 0.5773502692
The six trigonometric functions of a real angle are the complete set of signed ratios on the unit circle: sine and cosine are the coordinates of the point at that angle, tangent is their quotient, and cosecant, secant, and cotangent are the three reciprocals. Almost every programming library evaluates those functions on a radian argument, so a student, robotics script, or graphics pipeline that already holds theta in radians should not convert to degrees first or every ratio comes out wrong. This trigonometric functions calculator accepts one finite angle in radians and returns all six values together. Identities you can pin in tests include sine of pi over six equals one half, tangent of pi over four equals one, and secant of pi over three equals two.
What the six trigonometric functions return and why the argument is in radians
Trigonometry maps one real angle to six signed ratios on the unit circle. Sine is the vertical coordinate of the point at that angle, cosine is the horizontal coordinate, tangent is the slope of the ray, which is exactly sine divided by cosine whenever cosine is not zero, cosecant is one over sine, secant is one over cosine, and cotangent is cosine divided by sine.
Those six names are what people type when they search for trigonometric functions, not a short trig calculator that stops at the three primary values and not a sine-only degrees tool. Mainstream runtimes implement the analytic sine and cosine of a real argument in radians, so a caller who already holds theta as pi over six, one, or minus pi over four should send that number as-is.
Converting to degrees first and then forgetting to say so is the most common classroom bug: sine of thirty radians is about minus zero point nine eight eight, not one half. This landing is the radians-native six-function table. After evaluation, sine and cosine always land in the closed interval from minus one to one; a result outside that interval is floating-point noise and this implementation clamps it.
Special angles have textbook exact values that students expect without a trail of nines: sine of pi over six is one half, cosine of pi over three is one half, tangent of pi over four is one, and the three reciprocals follow from those identities. Negative angles are legal and use oddness for sine, tangent, cosecant, and cotangent and evenness for cosine and secant.
Angles larger than a full turn remain valid; they are reduced modulo two pi only when matching those special values, while the echoed input keeps the original measure. Related searches such as a three-function trig calculator that never emits the reciprocals, a sine-of-degrees page, or a degrees-or-radians six-function table that defaults to degrees, are different intents. Here the contract is all six trigonometric functions of one finite radian angle.
Where a six-function radian calculator earns its keep
Classrooms and tutoring platforms use a trigonometric functions calculator so learners can check the complete table for an arbitrary classroom angle in radians without mixing units or copying a calculator mode they cannot see. Exam generators need a deterministic oracle that marks non-finite tokens as invalid instead of silently returning NaN, which many language runtimes do and which a buggy grader might treat as a number. Failed validation does not consume balance.
Keep requests small, pin the rounded sextuple in tests, and you get reproducible trigonometric functions for homework, LMS plugins, and edge workers without standing up symbolic math infrastructure of your own.
Related searches such as inverse sine, a unit-circle explorer, a hyperbolic tangent, or a three-function degrees calculator, are different problems and are served by other pages; this contract stays on the six trigonometric functions of one finite radian angle so the landing matches what a student actually types when the question is simply the trigonometric functions of this many radians.
What you can do with it
Classroom and homework checks
Let students submit one angle in radians and verify all six trigonometric functions together, including the identities sin(π/6)=1/2, tan(π/4)=1, and sec(π/3)=2.
Exam and LMS auto-grading
Mark trigonometric-function items with a deterministic oracle that rejects non-finite angles and singularities, and never invents Infinity or NaN as a successful table.
Radian headings in robotics and simulation
Turn a radian heading into the six unit-circle ratios for a rise, a sprite offset, or a resolved force without importing a full computer-algebra library.
FAQ
Is the angle in degrees or radians?
Radians only. Library sine and cosine are analytic functions of a real radian argument. Convert degrees first, or use a degrees-capable calculator, instead of sending a degree number here.
What happens if the angle is not a finite number?
Missing fields, empty strings, non-numeric text, NaN, and Infinity are rejected and are not billed.
Why are 0 and π/2 rejected?
At 0 (and every integer multiple of π) sine is zero, so cosecant and cotangent are undefined. At π/2 (and every odd multiple) cosine is zero, so tangent and secant are undefined. Those angles fail rather than returning Infinity.
What are the six trigonometric values at π/6?
Sine 1/2, cosine √3/2, tangent 1/√3, cosecant 2, secant 2/√3 and cotangent √3. The three reciprocals are simply one over sine, one over cosine and cosine over sine.
How do I use an angle I only have in degrees?
Convert it to radians before entering it, because this calculator reads every angle as radians. Typing 30 here means 30 radians, whose sine is about −0.988, not the 1/2 you would expect from 30°.
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/trigonometric-functions \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"angle":0.523598775598}'const res = await fetch("https://api.kit.forhosting.com/math/trigonometric-functions", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"angle": 0.523598775598
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/trigonometric-functions",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"angle": 0.523598775598
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/trigonometric-functions", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"angle":0.523598775598}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"angle":0.523598775598}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/trigonometric-functions", 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": 0.523598775598
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.trigonometric_functions",
"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_abs_angle | 1000000000000 |
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
Non-finite tokens such as NaN and Infinity fail with invalid_input so a homework script never treats a missing table as a successful zero. Because cosecant and cotangent blow up at integer multiples of pi, and tangent and secant blow up at odd multiples of pi over two, this six-function contract rejects those angles instead of returning Infinity, which JSON cannot represent.
Input contract, singularities, and the response shape
There is no extra unit switch: this landing is the radians path on purpose, so a caller who holds degrees should convert them first or use a degrees-capable calculator instead of hoping a hidden flag will reinterpret the number. All six functions are returned only when every one of them is defined. If sine is zero, at integer multiples of pi, cosecant and cotangent are undefined. If cosine is zero, at odd multiples of pi over two, tangent and secant are undefined. On success the body echoes the cleaned angle in radians and returns sine, cosine, tangent, cosecant, secant, and cotangent as the six results. Example: angle pi over six yields sine one half, cosine root three over two, tangent one over root three, cosecant two, secant two over root three, and cotangent root three. POST a JSON body with the required field angle, a finite real number of radians. The field may arrive as a JSON number or as a numeric string in decimal or scientific notation; aliases theta, x, and radians are accepted when angle is omitted. Missing fields, empty strings, non-numeric text, NaN, and Infinity fail with invalid_input naming the field and are not billed. Those singularities fail with invalid_input rather than returning Infinity, which JSON cannot represent and which a homework script would treat as a number. A formula string restates the six identities, a compact expression naming all six of the input, and a one-line message complete the payload. Pair the payload with your own labels; this endpoint does not emit plots, a degree conversion table, or a step-by-step lesson beyond the formula field.
Where a six-function radian calculator earns its keep
Robotics, simulation, and game scripts already store headings in radians and still need opposite-over-hypotenuse, adjacent-over-hypotenuse, opposite-over-adjacent, and the three reciprocals for a rise, a sprite offset, or a resolved force; calling a tiny parse API avoids pulling a computer-algebra system for six evaluations.
Technical questions
How do I compute all six trigonometric functions of an angle?
Send a finite number of radians in the field angle (aliases theta, x, radians). The API returns sine, cosine, tangent, cosecant, secant, and cotangent together.
What does the trigonometric functions calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.