Math / Helper companion / Radians & models
Uncut · book companion
Kurs 4 radians & models — tangent via prior climb
When the turn uses a rope length
Degrees grew from sky-counting. Another measure grows from the circle itself: lay a rope along the rim equal to the radius, and the central angle is one radian. That rope language unlocks clean formulas, warns your calculator (DEG vs RAD), explains why tan rises between walls that never touch, and lets a harbour depth model predict when the water hits a flood line. (radianer · tangens · trigonometriska modeller)
Rebuild the turn from a rope on the rim — then convert and predict.
Builds on: full turn = 360° from the circle; tip coordinates from unit-circle climb.
Tool gained: radians = arc/radius; convert with 180° = π rad.
- Principle. A turn can be measured by how much rim rope it covers compared with the radius — not only by 360 equal ticks. When arc length equals the radius, the central angle is one radian. A full spin is 2π rad because the whole rim is 2πr.
- Model. Angle (rad) = arc / radius. Convert with 180° = π rad (× π/180 or × 180/π).
- Prediction / Answer. Convert a degree dial to radians before you trust a calculator in RAD mode — so two instruments name the same turn.
Tangent in models — uses prior tangent climb.
Builds on: Height ÷ run (tan = sin/cos, walls, P = 180°/k). Not a brand-new dump inside radians.
Tool gained: apply tan(kx) period/walls inside a modelling job (with optional radian dial later).
- Principle. Modelling jobs reuse the tangent tool already earned: steep climbs between walls where cos vanishes.
- Model. y = tan(kx), P = 180°/k; walls where cos(kx) = 0 — same as the tangent companion.
- Prediction / Answer. Space cutouts or phase-lock a dial with P — transfer, do not re-derive tan from scratch here.
Harbour tide model.
Builds on: wave model + meetings (trig equations) + radian argument tool above.
Tool gained: h(t) = A sin(kt)+B with P = 2π/k (radians) for depth / flood-line jobs.
- Principle. Harbour depth rises and falls on a repeating clock — the same “height that returns” idea, now in metres and hours.
- Model. h(t) = A sin(kt) + B (t in hours). With a radian argument, period P = 2π/k.
- Prediction / Answer. Plug in t for depth at a clock time; find where h(t) meets flood line L so a ferry can dock.
1. When the turn uses a rope length
A common book move: “1 radian ≈ 57.3°” as a fact to memorise, with no picture of why. Our rewrite: fix a circle of radius r. Mark an arc on the rim whose length is also r. The central angle that sits under that arc is defined as 1 radian. Double the arc → 2 rad; triple → 3 rad. The whole rim is 2πr, so one full turn is 2π radians — the same full turn degrees call 360°.
Rope rule: angle in radians = (arc length) / (radius). Full turn: 360° = 2π rad; half turn: 180° = π rad. Convert: degrees → radians multiply by π/180; radians → degrees multiply by 180/π. About 1° ≈ 0.0175 rad; 1 rad ≈ 57.3°.
Notation trap: if you see sin(2) with no degree mark, the 2 is almost always radians. Degrees need the ° symbol: sin(2°). Always check whether your calculator or app is in DEG or RAD before you trust a readout. (grader · radianer)
2. Hands-on — degree ↔ radian converter
Drag either dial. The other updates so both name the same turn. Watch 180° lock to π, and 360° lock to 2π.
180° = π rad ≈ 3.1416
Internal: rad slider is stored as hundredths (314 → 3.14). Exact: deg × π/180 = rad.
3. Tangent — steep climbs between walls
On the unit circle, tan x = (height)/(sideways) = sin x / cos x. Whenever cos x = 0, the sideways run vanishes and tan is undefined — the graph shoots toward ±∞ and never crosses those vertical walls (asymptotes). For y = tan x those walls sit at 90° + n·180° (n = 0, ±1, ±2, …). Between two neighbouring walls the curve climbs through (0,0) once — so the period is 180°, half a full spin. (tangens · lodräta asymptoter · period)
Period of tangent: for y = tan(kx), P = 180°/k (degrees). Larger |k| packs the walls tighter. Steeper stretch: y = A tan(x) keeps the same period as tan x but climbs faster between the walls. Shifts work like sine: y = tan x + B lifts; y = tan(x + v) slides (v>0 → left in the usual book form).
| x | tan x (approx) |
|---|---|
| −60° | −1.73 |
| −30° | −0.58 |
| 0° | 0 |
| 30° | 0.58 |
| 60° | 1.73 |
| ±90° | undefined |
Example — solve tan 2x = 0.85 (degrees)
Example — beacon phase: tan 2x = 0.85 on 0°…360°. Principal: 2x ≈ 40.4° + n·180°. Divide by 2: x ≈ 20.2° + n·90°. In the window: about 20°, 110°, 200°, 290° (whole degrees). Same shape as a naked “solve tan 2x = …” drill — we gave the dial a job (phase lock every quarter period).
Weapon: general solution for tan θ = a is θ = arctan(a) + n·180°, then unwind θ = 2x.
4. Harbour model — depth that returns
Sine and cosine name repeating jobs in the world: temperature, circular motion, waves. A harbour depth that rises and falls with the tide is a classic model.
Tide model: h(t) = A sin(kt) + B. Here t is hours after a reference clock, h is depth in metres, |A| is half the swing, B is the midline (mean depth), and the period is P = 2π/k when the argument is in radians. Predict depth by plugging in t; predict clock times by finding where h(t) meets a flood line L.
Worked harbour (reframed)
Harbour depth: h(t) = 2.8 sin(0.52 t) + 6.0, with t hours after 09:00. At 09:00, t=0 → h=6.0 m. At 13:00, t=4 → h ≈ 2.8 sin(2.08) + 6.0 ≈ 8.5 m. Period P = 2π/0.52 ≈ 12 hours — one full tide cycle about twice a day. Flood line L = 4.0 m meets the wave near t ≈ 8 h and 10 h after 09:00 (17:00 and 19:00), then again about 12 h later.
Core pedagogy unchanged from the cut harbour story: plug in t, read P = 2π/k, find meetings with a depth line. Numbers varied slightly on purpose.
5. Hands-on — tide vs flood line
Optional graph: h(t) = 2.8 sin(0.52 t) + 6.0 over 0…24 hours. Drag the flood line L and watch how many meetings appear in one day.
L = 4.0 m · meetings in 0…24 h ≈ 4
Moving L also counts toward hands-on credit if you have not yet used the converter.
6. Fake cut tasks → real jobs
Fake
“Convert 45° to radians.” Symbol in, symbol out — no instrument that needs both dials.
Real — two instruments, one turn
A dock winch reports shaft angle in degrees; a control script expects radians. 45° = π/4 rad. You convert so the script and the winch name the same turn — otherwise the arm overshoots.
How the fake becomes real
Same × π/180. The real job supplies two devices that must agree.
Fake
“State the period of y = tan(3x).” Coefficient in, formula out.
Real — scan walls on a rotating mirror
A lidar mirror’s scan intensity follows a tan-like rise three times per half-turn of interest: model ∝ tan(3x). P = 180°/3 = 60°. Designers space safety cutouts between those walls — same P = 180°/k, with a machine that can jam if the walls are ignored.
How the fake becomes real
Weapon P = 180°/k for tan(kx). The stake is keeping the scan from hitting an asymptote blind spot.
Fake
“Find P for h(t) = 2.8 sin(0.52t) + 6.” No harbour, no clock.
Real — when can the ferry dock?
Ferry needs depth above 4.0 m. With h(t) = 2.8 sin(0.52t) + 6.0 and P ≈ 12 h, you know how often the flood line is cleared each day. Schedule around the meetings — same 2π/k, now a timetable.
How the fake becomes real
Period tool plus flood-line crossings. The ferry supplies the why.
Fake
“sin(2) — what mode?” A gotcha with no story.
Real — DEG/RAD mismatch on the dock
A crew runs sin(2) meaning 2°, but the tablet is in RAD. sin(2 rad) ≈ 0.91; sin(2°) ≈ 0.035. The hoist load estimate is wildly wrong until someone flips the mode. Notation without ° means radians in math writing — check the tool.
How the fake becomes real
Same warning as the chapter tip; the stake is a wrong load on a cable.
Gate — four, all required
Hands-on (move the converter or the flood line) is 20. Each right gate item is 20. Rebuild and predict — do not only recite a conversion table.
1. One radian is best read as…
2. Why does y = tan x have period 180°, not 360°?
3. Harbour h(t) = 2.8 sin(0.52 t) + 6.0 (t in hours, argument in radians). Predict the period.
4. You raise the flood line L from 4.0 m toward the crest (~8.8 m) on that harbour wave. Predict the meetings in one day.