Trigonometry OS (A-Math) — Full Operating System Page

Unified Almost-Code (Z0–Z3) + Failure Atlas + Sensors + Repairs + Domain/General-Solution Governance v0.2

PAGE-ID: EDK-LAN-AMTH-TRIG-OS-v0.2
Role: LAN.TopicOS + IdentityEngine + EquationPipeline + DomainControl + TrapGovernance
Purpose: make Trigonometry reliable: identities, equations, domains, and general solutions without trap collapse under time.

0) Definition Lock

Trigonometry OS := the capability system for:

  • manipulating trig identities cleanly
  • solving trig equations correctly (with domain + general solutions)
  • selecting correct identities quickly
  • avoiding domain and form traps under exam load

P3 Trigonometry condition:

Can simplify identities, solve trig equations with correct general solutions and domains, and survive timed variation with low trap activation.

1) Trigonometry OS Scope (O-Level A-Math)

Core families:

  • fundamental identities (sin²x + cos²x = 1, etc.)
  • converting between tan, sin, cos
  • solving basic trig equations
  • general solutions (0°–360° or 0–2π depending on mode)
  • simple trig identities (prove/simplify)
  • equation solving with identities
  • domain restriction awareness

2) Z0–Z3 Directory (Skills → Failure Modes → Sensors → Repairs)

Z0 — Kernel Atoms (Identity + Algebra Cleanliness)

Node T0.1 — Fundamental Identities Fluency

NodeID: TRIG-Z0-ID-FUND
Skills:

  • sin²x + cos²x = 1
  • tan x = sin x / cos x
  • 1 + tan²x = sec²x (if in scope)

Failure modes: forget identity; substitutes wrongly; algebra errors (B1/B2)
Sensors: Concept Sensor; Error Recurrence
Repairs:

  • Identity micro-drills (no solving)
  • “convert to one trig function” discipline
  • retest ≤ 7 days

Node T0.2 — Algebra Kernel for Trig

NodeID: TRIG-Z0-ALG-KRN
Skills: factorising trig expressions; solving quadratic-in-sin/cos form
Failure modes: algebra swamp; sign errors; cannot factorise trig expressions (B2)
Sensors: Foundation Fluency; Messiness
Repairs:

  • A-Math Fluency Kernel daily
  • “rewrite as quadratic in sin/cos” rule

Trap: TR1 sign trap.

Node T0.3 — Calculator Mode Discipline (Degree/Radian)

NodeID: TRIG-Z0-CALC-MODE
Skills: correct angle mode; interpreting calculator outputs
Failure modes: wrong mode; wrong answer scale (B5/TR4)
Sensors: Error Recurrence; Trap frequency
Repairs:

  • mode-check rule before solving
  • post-solve magnitude sanity check

Z1 — Method Signals + Domain Governance

Node T1.1 — Solve Basic Trig Equations (Single Function)

NodeID: TRIG-Z1-SOLVE-BASIC
Skills:

  • sin x = k
  • cos x = k
  • tan x = k

Failure modes: missing second solution; wrong quadrant logic (B1/B3/TR4)
Sensors: Method Selection; Trap activation rate
Repairs:

  • quadrant sketch drills
  • “two angles in 0–360° unless boundary case” rule
  • general solution template

Node T1.2 — Quadratic-in-Trig Equations

NodeID: TRIG-Z1-QUAD-FORM
Skills: rewrite as quadratic in sin/cos; factorise; solve
Failure modes: forget substitution; lose solutions; algebra errors (B3/B2)
Sensors: Method Selection; Algebra Kernel
Repairs:

  • substitution pipeline drills
  • retest within 7 days

Node T1.3 — General Solution Governance

NodeID: TRIG-Z1-GEN-SOL
Skills:

  • express solutions with 360°n / 2πn
  • restrict to required domain

Failure modes: gives only principal value; wrong general form; ignores domain (TR4)
Sensors: Trap activation; Error Recurrence
Repairs:

  • “solution template” rule:
  • first find base angles
  • then apply general form
  • then restrict domain

Z2 — Transfer + Identity Manipulation Chains

Node T2.1 — Identity Simplification (Prove/Rewrite)

NodeID: TRIG-Z2-IDENTITY
Skills: rewrite LHS to RHS; use identities strategically
Failure modes: random substitution; algebra mess; circular proof (B3/B2)
Sensors: Bandwidth (structure recognition); Messiness
Repairs:

  • signal drills: “convert to sin/cos only”
  • identity chain discipline (one rewrite per line)

Node T2.2 — Mixed Trig Equations (Multi-step)

NodeID: TRIG-Z2-MIX-EQ
Skills: combine identity + algebra + solving
Failure modes: forget domain; drop solutions; sign errors (B2/B3/TR2)
Sensors: Transfer; Trap frequency
Repairs:

  • pipeline template:
  1. rewrite
  2. simplify
  3. solve
  4. apply general solution
  5. restrict domain

Node T2.3 — Variation Ladder (Form Change Survivability)

NodeID: TRIG-Z2-TRANSFER-VAR
Failure modes: can do drills; fails on wording changes; freezes on identity form (B3 hidden B1)
Sensors: Transfer/Variation
Repairs:

  • T0→T4 ladder
  • method-signal annotation before solving

Z3 — Timed Stability + Trap Audit (Exam Reliability)

Node T3.1 — Timed Trig Mini-Paper

NodeID: TRIG-Z3-TIMED
Failure modes: slow identity manipulation; domain forgotten under pressure; careless spikes (B6→B5)
Sensors: Pace Budget; DriftRisk
Repairs:

  • timed minis (smaller scope)
  • skip/return policy
  • final 5–10 min domain audit

Node T3.2 — Trig Trap Audit Checklist

NodeID: TRIG-Z3-TRAP-AUDIT

Common traps:

  • TR1 sign trap in algebra
  • TR4 missing second angle
  • TR4 missing general solution
  • TR2 domain restriction ignored
  • calculator mode error

Audit stack:

  1. Did you find all angles in required domain?
  2. Did you state general solution correctly?
  3. Did you apply domain restriction properly?
  4. Calculator mode correct?
  5. Does answer match what question asked?

3) Trig Sensors (Topic-specific pack)

Minimum sensors:

  • Identity recall accuracy
  • Algebra cleanliness
  • Method selection speed
  • Domain/general-solution correctness rate
  • Pace stability
  • Trap activation frequency

4) Trig Labs (Topic-specific drill set)

  • Lab-T1: Identity micro drills
  • Lab-T2: Solve-basic-equation quadrant drills
  • Lab-T3: Quadratic-in-trig substitution lab
  • Lab-T4: Identity chain rewrite lab
  • Lab-T5: General-solution template lab
  • Lab-T6: Timed mini + domain audit routine

5) Trig Gauge (Topic Scorecard)

Weekly:

  • Precision __/5
  • Bandwidth __/5
  • Speed __/5
  • DriftRisk Low/Med/High
  • Dominant bucket B1–B6

Readiness targets:

  • Precision ≥ 3
  • Bandwidth ≥ 3
  • Speed ≥ 3
  • DriftRisk not High
  • Domain/general-solution errors trending down

6) Operator Checklist (Copy-Paste Block)

[Trigonometry OS Operator Checklist v0.2]
1) Rewrite to sin/cos when stuck
2) Solve base equation first
3) Apply general solution form
4) Restrict to required domain
5) Audit for missing angles or mode errors
6) Retest within 7 days if trap repeats

Start here if you want the full sequence:

Vocabulary OS Series Index:
https://edukatesg.com/vocabulary-os-series-index/

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