Math / Helper companion / Unit circle
Uncut · book companion · ~10–24
Kurs 4 Kap 1 start — measure → turn table → symmetries
From roof slope to turn table
You can measure a roof’s height from its run and slope — until the job asks for turns past 90°, or exact values without a calculator. Climb: ratios that measure the unmeasurable → the unit circle as a turn table → symmetries as tools for exact values and basic equations. Span grows by at most one each step. (enhetscirkeln · sinus · cosinus · exakta värden)
V1 — Measure the unmeasurable with ratios.
Brand-new (or Builds on: right-triangle quest) · span 0–1
Tool gained: sin / cos / tan as opposite÷hypotenuse, adjacent÷hypotenuse, opposite÷adjacent in a right triangle.
- Principle. A height you cannot climb can still be found if you know a run and an angle — similar right triangles share side ratios.
- Model. sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj for acute θ in a right triangle.
- Prediction / Answer. A 12 m guy wire at 35° to the ground: height = 12 sin 35°. Same ratios for a new roof or mast job.
V2 — Circle as turn table.
Builds on: sin/cos as triangle ratios (V1 tool).
Tool gained: on the unit circle, (cos θ, sin θ) is the tip after turn θ — works past 90°.
- Principle. Triangle ratios stop at acute angles. A turn table of radius 1 keeps the same “height” and “sideways” meanings for any turn — including past a right angle.
- Model. Unit circle: tip at angle θ has coordinates (cos θ, sin θ). Signs flip by quadrant; hyp = 1 so ratios become coordinates.
- Prediction / Answer. At 120°, cos is negative (left), sin still positive (up). Transfer a mast angle past 90° without inventing a new trig.
V3 — Symmetries as tools.
Builds on: unit-circle tip (V2 tool).
Tool gained: quadrant signs + co-function / reference-angle symmetries for exact values and basic eqs.
- Principle. The turn table is symmetric: reflections and 180° flips reuse the same tip lengths with sign changes. You do not need a new table for every angle.
- Model. Reference angle in Q1; apply quadrant signs. Exact tips: 0°, 30°, 45°, 60°, 90° (and symmetries). Basic: sin θ = k → θ = arcsin k + 360°n or 180° − arcsin k + 360°n (degrees).
- Prediction / Answer. Exact: cos 150° = −√3/2. Solve sin θ = 1/2 on 0°…360° → 30° and 150°. Same symmetry tool, with a job.
1. Poor dump → rewrite
Book dump: paste SOHCAHTOA, then a unit-circle picture, then a table of exact values — three topics taped together. Our rewrite: ratios earn the acute job; the circle reuses those ratios as coordinates so turns past 90° still mean height/sideways; symmetries reuse one tip to fill exact values and solve basic equations.
Climb weapons: triangle ratios → unit-circle tip (cos, sin) → reference angle + quadrant signs. Exact values are readouts of symmetries, not wallpaper.
2. Hands-on — spin the tip
Drag angle θ. Read (cos θ, sin θ) and the reference angle. Watch signs flip by quadrant.
cos 30° ≈ 0.866 · sin 30° = 0.5 · ref 30° · Q1
3. Exact values from symmetries
| θ | sin | cos |
|---|---|---|
| 0° | 0 | 1 |
| 30° | 1/2 | √3/2 |
| 45° | √2/2 | √2/2 |
| 60° | √3/2 | 1/2 |
| 90° | 1 | 0 |
Table is a readout. Symmetry + quadrant signs extend it — e.g. sin 150° = sin 30°.
Fake
“Memorise sin 150°.” No why.
Real
A solar tracker needs exact tip height at 150° from the noon mark. Reference 30°, Q2 ⇒ sin positive = 1/2. Same symmetry tool as the table — now a machine setting.
Gate — four, all required
Hands-on (move θ) is 20. Gates: reconstruct, transfer, predict.
1. Why move from triangle ratios to the unit circle?
2. On the unit circle, cos 120° is…
3. Predict solutions of sin θ = √2/2 on 0°…360°.
4. Span check: jumping from “acute SOHCAHTOA only” straight to “solve A sin(kx+v)=B with identities” would…