Math / Helper companion / Kap 1 wrap
Uncut · book companion · ~65–80
Integration climb — spend every Kap 1 Tool on one job chain
Kap 1 wrap — mix the trig tools
This is not a drill dump. Climb by spending Tools you already earned: tip coordinates and formulas for a clean identity job → wave model + meeting for a flood-line clock → tangent walls + radian rope on the same harbour dial. Span grows by at most one Tool family per Vortex. Fake→Real bank forces transfer across tools. (sammanfattning · blandad · verktyg överföring)
V1 — Unit tip + formula Tool on one identity job.
Builds on: unit-circle tip (cos θ, sin θ) + formula Tools (unit / addition). Not a new identity chapter.
Tool gained: pick the right prior formula for a job — e.g. cos²θ + sin²θ = 1 or cos(A−B) — from the tip picture, not from chant.
- Principle. A Kap 1 “summary” is useless unless the tip and the formula are the same object: coordinates on the unit circle are the reasons the identities hold.
- Model. For a point (cos θ, sin θ) on the unit circle: cos²θ + sin²θ = 1. Addition formulas come from composing two turns.
- Prediction / Answer. A robot arm at angle θ needs |tip| = 1 on the unit map. Check cos²θ + sin²θ before trusting a sensor dump — transfer of the unit Tool into QA.
V2 — Wave model + meeting (functions + equations).
Builds on: V1 formula spend + wave model Tool (functions) + meeting Tool (equations). Span +1: now the identity sits inside a timed wave job.
Tool gained: write h(t)=A sin(kt)+B (or cos), then solve h(t)=L with general solution + period — one chain across functions and equations.
- Principle. A height that returns is a wave model; a flood line is a meeting. Kap 1 wrap means you do both on the same harbour, not separate drill sheets.
- Model. h(t)=A sin(kt)+B; solve sin(kt)=(L−B)/A; add 2πn/k (radian dial) or 360°/k (degree dial).
- Prediction / Answer. Tide A=1.2 m, B=2.0 m, k=π/6 per hour (rad). Flood L=2.6 → sin(kt)=0.5 → first hits at t=1 h, then every 12 h family — transfer, not a naked solve-for-x.
V3 — Tangent walls + radian rope on the same dial.
Builds on: V2 wave/meeting chain + tangent Tool (height÷run, walls) + radian Tool. Span +1: same job may need a steep-climb read between walls.
Tool gained: Kap 1 transfer pack — switch unit/formula ↔ wave/meeting ↔ tan/radian on one real dial without dropping Tools.
- Principle. Some jobs are steep between forbidden walls (tan), timed on a rope dial (radians), and still meet a line (equations). Wrap = choose which Tool speaks.
- Model. y=tan(kt) with walls where cos(kt)=0; or convert the meeting angle to radians before a RAD-mode calculator.
- Prediction / Answer. Phase-lock a cutout with P=π/k (rad) between walls, then check a sin meeting on the same clock — two Tools, one harbour board.
1. Poor dump → rewrite
Book dump: twenty mixed drills with no Tool labels — “simplify, solve, convert.” Our rewrite: every item must name which prior Tool it spends, and the Fake→Real bank shows the job that made the Tool worth keeping.
Weapons: unit/formula pick → wave+meeting chain → tan/radian switch. Kap 1 closed as an integration climb; Kap 2 opens on derivative as slope.
2. Hands-on — harbour dial (A, k, L)
h(t)=A sin(kt)+2. Drag A, k, flood L. Read first meeting in the positive family (radian dial).
…
3. Fake → real bank (transfer across Tools)
Fake A
“Prove cos²θ+sin²θ=1.” Identity only.
Real A
Sensor QA on a unit-mapped arm: reject packets where cos²+sin² ≠ 1 within tolerance — same unit Tool, now a reject gate.
Fake B
“Solve sin x = 1/2.” No clock, no job.
Real B
Ferry dock when tide h(t)=1.2 sin(πt/6)+2 hits 2.6 m — meeting Tool on a 12 h family, not a naked x.
Fake C
“Convert 90° to radians.” Conversion drill.
Real C
Calculator in RAD mode must see π/2 before you trust sin of a “right angle” cut — rope Tool stops a degree/radian mix-up on the shop floor.
Gate — four, all required (transfer)
1. Reconstruct: cos²θ+sin²θ=1 is true because…
2. Why does a flood-line job need both functions and equations Tools?
3. Predict: h(t)=1.2 sin(πt/6)+2, L=2.6. First t>0 with sin=1/2?
4. Transfer: RAD-mode calculator + a “90° cut” — which Tool?