Tangent double-angle calculator
Double an angle's tangent without knowing the angle: a tangent of 0.5 gives a numerator of 1 and a denominator of 0.75, so tan(2θ) is 1 divided by 0.75. Values of 1 and −1 are flagged as undefined. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Tan theta
- 0.5
- Numerator
- 1
- Denominator
- 0.75
- Tan double theta
- 1.3333333333
The tangent double-angle calculator finds tan(2θ) when you already know tan(θ). It substitutes the supplied tangent into 2 tan(θ) divided by 1 minus tan squared(θ), reports the numerator and denominator, and returns the resulting tangent. If the denominator is zero, the doubled-angle tangent is undefined and the calculator reports an input error instead of presenting an enormous or misleading number. This focused input format is useful for algebra exercises, identity checks, and applications where the original angle itself is unknown.
Use a known tangent without reconstructing the angle
The calculator does not need θ in degrees or radians because the double-angle identity works directly with the tangent value. For example, if tan(θ) is 0.5, the numerator becomes 1 and the denominator becomes 0.75, so tan(2θ) is 1 divided by 0.75. This is especially convenient when a problem gives a ratio from a right triangle or an algebraic condition but never supplies the angle itself.
Decimal values and finite numeric strings are accepted, while blank, nonnumeric, infinite, and NaN inputs are rejected. Because tangent is periodic, one tangent value can correspond to many possible angles, but all of those angles produce the same value under this identity whenever the doubled-angle tangent is defined.
Understand the identity and the undefined case
The calculation uses tan(2θ) = 2 tan(θ) / (1 - tan²(θ)). If t represents tan(θ), the operation is simply 2t divided by 1 - t². The numerator preserves the sign of t, while the denominator can be positive, zero, or negative.
When t is 1 or -1, t² is 1 and the denominator is exactly zero. Values close to 1 or -1 are still mathematically defined, although their results may be very large because the denominator is small. The calculator does not silently round such inputs into the undefined case.
It evaluates the finite number supplied by the caller and rejects the result only when the computed denominator is exactly zero or the arithmetic cannot produce a finite output. This distinction preserves legitimate near-asymptote calculations while making the true singularity explicit.
What you can do with it
Check trigonometry homework
Verify a tangent double-angle substitution and compare the numerator, denominator, and final value with handwritten work.
Work from a triangle ratio
Use a known opposite-to-adjacent ratio as tan(θ) without first finding θ with an inverse function.
Detect singular results
Catch tan(θ) values of 1 or -1, where the identity denominator is zero and tan(2θ) is undefined.
FAQ
What input does the calculator require?
You do not need to provide θ itself.
Which formula is used?
It uses tan(2θ) = 2 tan(θ) / (1 - tan²(θ)).
When is the result undefined?
The result is undefined when tan(θ) is 1 or -1, because the denominator 1 - tan²(θ) is zero.
Does the input use degrees or radians?
Neither unit is needed. The input is already a tangent value, which is a unitless ratio.
Is tan(2θ) just twice tan θ?
No. Doubling the tangent only gives the numerator of the formula; you still divide by one minus the tangent squared. With a tangent of 0.5, twice it is 1, but the answer is 1 divided by 0.75.
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/tan-double-angle \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"tan_theta":0.5}'const res = await fetch("https://api.kit.forhosting.com/trig/tan-double-angle", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"tan_theta": 0.5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/trig/tan-double-angle",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"tan_theta": 0.5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/trig/tan-double-angle", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"tan_theta":0.5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"tan_theta":0.5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/trig/tan-double-angle", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"tan_theta": 0.5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "trig.tan_double_angle",
"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
Use a known tangent without reconstructing the angle
Enter the known value of tan(θ) in the tan_theta field. The response echoes the normalized tangent, displays both intermediate parts of the fraction, and names the formula used. Those intermediate values make it easier to compare your handwritten substitution with the computed result and find a sign or squaring mistake.
Understand the identity and the undefined case
Division by zero does not define a real tangent value, so the capability flags the output as undefined through a clear invalid_input error. This corresponds to doubled angles whose cosine is zero, such as 90 degrees modulo 180 degrees.
Interpret and verify the returned result
The response includes numerator and denominator so you can verify every substitution. A negative result can arise from a negative numerator or from a negative denominator; for instance, a positive tangent larger than 1 produces a positive numerator but a negative denominator. The undefined field is false on every successful response, because undefined cases are returned as errors rather than ordinary numeric records. Read tan_double_theta as the value of tan(2θ), not as the doubled value of tan(θ). Those expressions are generally different: doubling the tangent gives only the formula numerator, and the denominator correction is essential. This makes automated use straightforward: successful calls always contain a finite result, while callers can handle invalid_input separately. The calculation is deterministic, has no network dependency, and uses the same pure solver for the API and browser experience. It works well as a focused identity checker, a step in a larger trigonometry workflow, or a teaching aid for exploring how tangent changes when an angle doubles. API requests use the published base price of $0.002; the page can also execute the calculation locally in the browser.
Technical questions
How much does an API calculation cost?
Each API request uses the published base price of $0.002. Browser execution is also available on the page.