Math / Helper companion / Trig formulas
Uncut · book companion · ~25–38
Each formula earned — no dump
Formulas earned as tools
A wall of identities is memorization. Climb instead: Pythagoras on the unit tip → add turns by composing tips → double a turn as a special case. Each formula is the Tool gained for the next job. Span steps Δ≤1. (trigonometriska formler · additionsformler · dubbla vinkeln)
V1 — Unit identity from the tip.
Builds on: unit-circle tip (cos θ, sin θ) from trig-unit-circle.
Tool gained: cos²θ + sin²θ = 1 (and rearrangements).
- Principle. The tip sits on a circle of radius 1, so its sideways and height already satisfy Pythagoras.
- Model. cos²θ + sin²θ = 1. Then 1 + tan²θ = 1/cos²θ when cos ≠ 0.
- Prediction / Answer. If sin θ = 3/5 and θ in Q1, cos θ = 4/5. Check a sensor that reports only one coordinate.
V2 — Adding turns.
Builds on: unit identity + tip coordinates (V1 / unit circle).
Tool gained: cos(a±b), sin(a±b) as composition of two tips.
- Principle. Turning by a then by b is one combined turn. The new tip’s coordinates must come from the two known tips — not a new table.
- Model. cos(a−b) = cos a cos b + sin a sin b; cos(a+b) = cos a cos b − sin a sin b; sin(a±b) = sin a cos b ± cos a sin b.
- Prediction / Answer. Exact: cos 75° = cos(45+30) = (√6 − √2)/4. Transfer: phase-shift a signal by a known angle.
V3 — Double angle as a special case.
Builds on: addition formulas (V2 tool).
Tool gained: sin 2θ, cos 2θ (and cos 2θ in three useful forms).
- Principle. Doubling a turn is adding the turn to itself — so addition formulas specialize; you do not memorize a separate cult of “double angle.”
- Model. sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ.
- Prediction / Answer. If cos θ = 0.6, cos 2θ = 2(0.6)² − 1 = −0.28. Pick the form that matches what you know (sin-only, cos-only, or mixed).
1. Poor dump → rewrite
Book dump: a page of framed identities to chant. Our rewrite: each identity is earned when a job needs it — Pythagoras on the tip, then compose two turns, then set the two turns equal.
Weapons in order: cos²+sin²=1 → addition/subtraction → double angle. Never leap to double angle without addition as the prior tool.
2. Hands-on — check addition
Pick a and b. Compare cos(a+b) from the formula vs cos of the sum (calculator check in degrees).
…
3. Fake → real
Fake
“State cos 2θ in three forms.” List with no job.
Real
A power meter gives sin θ only. You need cos 2θ for a torque estimate — use 1 − 2 sin²θ. Same identity, now a sensor transfer.
Gate — four, all required
1. Reconstruct: cos²θ + sin²θ = 1 comes from…
2. Why is double angle after addition in the climb?
3. Predict cos 15° using 45°−30°.
4. Transfer: you know only cos θ = 0.8. Best cos 2θ form?