Sign of trig function in quadrant calculator

Non-numeric quadrants outside one through four and unrecognized function names fail with invalid_input and are not charged.

What the ASTC rule means for trig signs by quadrant

Delivering the sign, a plus or minus symbol, a boolean, the quadrant label, and the positive-function list makes the result easy to paste into notes, assert in automated tests, or feed into a larger pipeline without reimplementing the ASTC table in every language you teach or deploy.

How this calculator validates input and applies the ASTC table

Function must be one of the six standard names sin, cos, tan, csc, sec, or cot; common full spellings such as sine, cosine, tangent, cosecant, secant, and cotangent are accepted and normalized, and matching is case-insensitive. Values outside that range, non-integers, and unparseable text are rejected before any table lookup runs. Edge cases are explicit: sine in quadrant two is positive with mnemonic word Students; cosine in quadrant two is negative; tangent in quadrant three is positive with mnemonic word Take; secant in quadrant four is positive; cotangent in quadrant one is positive because all six functions share that region. Send a JSON body with required fields function and quadrant. Empty strings, null, and unknown tokens fail with invalid_input so those failed calls are not charged. Quadrant must resolve to an integer in the closed set one through four; JSON numbers two and three work, as do string forms two, II, Q2, and quadrant three after a light prefix strip. When both fields are valid the engine looks up the set of functions that stay positive in that quadrant under the ASTC rule, tests membership of the chosen function, and builds a stable response: the normalized function name, numeric quadrant, roman quadrant_label, sign as the word positive or negative, sign_symbol as plus or minus, is_positive as a boolean, the full mnemonic string all students take calculus, the single mnemonic_word All, Students, Take, or Calculus for that quadrant, the ordered positive_in_quadrant list, rule ASTC, and method sign_of_trig_function.

Practical use, limits, pricing, and delivery channels

Use the browser tool when you want an immediate plus or minus while checking homework, labeling a unit-circle sketch, or confirming a sign chart for sine and cosine without installing a computer-algebra system. Results are fully deterministic: the same pair always yields the same classification. There is no language model, no external geometry service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials. Axial angles are outside the contract because open-quadrant sign charts do not assign a unique ASTC cell when the terminal side lies on an axis. Each successful request is one billable item at $0.002; failed validation does not consume balance. Accepted input is a recognized trig name and a quadrant from one to four; the response echoes the normalized function and quadrant and reports sign fields with method sign_of_trig_function. Wire the endpoint into unit-circle demos, contest answer checkers, worksheet generators that print ASTC tables, or floating pipelines where you already prefer explicit domain errors for bad payloads over opaque black-box widgets that cannot show why a request failed to parse. It does not replace a quadrant-of-angle calculator when you still need to discover which quadrant a degree or radian measure occupies, and it does not evaluate numeric sin or cos values—those are separate tools with different search intent and different outputs.