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Trigonometry Calculator
Evaluates the six trigonometric functions of an angle given in degrees or radians, with the reference angle and quadrant, and/or the inverse function (arcsin, arccos, arctan) of a value, returning the angle in degrees and radians.
When to use
You need sin, cos, tan or their reciprocals for an angle, the angle whose sine/cosine/tangent is a given value, or the reference angle and quadrant of an angle.
Do not use when: You want to solve a triangle from sides and angles (use right-triangle or triangle-solver) or only convert an angle between units (use angle).
Formula
θ_rad = θ_deg × π/180; tan = sin/cos, sec = 1/cos, csc = 1/sin, cot = cos/sin; reference angle = acute angle to the x-axis; arcsin, arccos, arctan return principal values
Angles that are whole degrees are evaluated exactly at the axes (sin 180° = 0, cos 90° = 0) so that the reciprocal functions report 'undefined' instead of a huge number; other values come from IEEE double arithmetic and are rounded to 6 decimals.
Inputs
| Parameter | Type | Unit | Required | Description |
|---|---|---|---|---|
angle | number | no | Angle to evaluate the forward functions at. Omit when only an inverse function is needed. Range: ≥ -1000000000, ≤ 1000000000 | |
unit | enum: degrees | radians | default degrees | Unit of the angle input. | |
value | number | no | Number whose arcsin, arccos or arctan is wanted (between −1 and 1 for arcsin/arccos). | |
inverse_function | enum: arcsin | arccos | arctan | default arcsin | Which inverse function to apply to value. |
Outputs
| Output | Type | Unit | Description |
|---|---|---|---|
angle_degrees | number | ° | The input angle in degrees. |
angle_radians | number | rad | The input angle in radians. |
normalized_degrees | number | ° | The equivalent angle in [0°, 360°). |
reference_angle_degrees | number | ° | Acute angle between the terminal side and the x-axis (0°–90°). |
quadrant | string | Quadrant I–IV of the terminal side, or the axis it lies on. | |
sin | number | Sine of the angle. | |
cos | number | Cosine of the angle. | |
tan | number | Tangent sin/cos; the string 'undefined' at 90° + k·180°. | |
sec | number | Secant 1/cos; 'undefined' where cos = 0. | |
csc | number | Cosecant 1/sin; 'undefined' where sin = 0. | |
cot | number | Cotangent cos/sin; 'undefined' where sin = 0. | |
inverse_angle_degrees | number | ° | inverse_function(value) in degrees: arcsin in [−90°, 90°], arccos in [0°, 180°], arctan in (−90°, 90°). Only when value is given. |
inverse_angle_radians | number | rad | The same principal value in radians. |
inverse_expression | string | e.g. 'arcsin(0.5) = 30°'. |
Example
30° and arcsin 0.5: {"angle":30,"unit":"degrees","value":0.5,"inverse_function":"arcsin"} → {"angle_degrees":30,"angle_radians":0.523599,"normalized_degrees":30,"reference_angle_degrees":30,"quadrant":"Quadrant I","sin":0.5,"cos":0.866025,"tan":0.57735,"sec":1.154701,"csc":2,"cot":1.732051,"inverse_angle_degrees":30,"inverse_angle_radians":0.523599,"inverse_expression":"arcsin(0.5) = 30°"}
90°: {"angle":90} → {"sin":1,"cos":0,"tan":"undefined","sec":"undefined","csc":1,"cot":0,"reference_angle_degrees":90,"quadrant":"on the positive y-axis (90°)","angle_radians":1.570796}
GET https://tttkmbb.com/api/v1/calculate/trigonometry?angle=30&unit=degrees&value=0.5&inverse_function=arcsin
Machine access
- API:
GET https://tttkmbb.com/api/v1/calculate/trigonometry(query parameters) orPOSTwith a JSON body{"inputs": {...}} - Schema: https://tttkmbb.com/api/v1/calculators/trigonometry · Markdown: https://tttkmbb.com/math/trigonometry.md · JSON definition: https://tttkmbb.com/math/trigonometry.json
- MCP: server
https://tttkmbb.com/mcp, toolrun_calculator with calculator_id="trigonometry" - OpenAPI operationId:
calculate_trigonometric_functions - Freshness:
static. Authentication: none. Rate limit: fair use (see rate limits).
Sources
FAQ
Why is tan 90° 'undefined' rather than a large number?
cos 90° is exactly 0, so tan = sin/cos has no value; the function has a vertical asymptote there. A plain calculator prints ~1.6 × 10^16 only because π/2 cannot be represented exactly.
Which value does arcsin return?
The principal value between −90° and 90°. Since sin is periodic, 30° and 150° both have sine 0.5; the second solution of sin x = v is 180° − arcsin v.
What is the reference angle for?
It is the acute angle whose function values equal those of the original angle up to sign, e.g. sin 150° = sin 30° and cos 150° = −cos 30°.
Related calculators
- Angle Converter — Convert between degrees, radians, gradians and turns.
- Right Triangle Calculator — Apply the ratios to the sides of a right triangle.
- Triangle Solver — Law of sines and cosines for general triangles.