Math / Helper companion / Trig functions
Uncut · book companion
Kurs 4 §1.3 ideas — our narrative
When the circle keeps turning
Day length, tides, a radio bob, a weather balloon tugging on a line — many jobs come back again and again. You need a function that repeats. The cleanest engine is still the Ring of Measure: spin a radius, read its height. That height, plotted against the turn, is the wave. (periodisk företeelse · sinus · enhetscirkel)
Rebuild the wave from the turning circle — do not memorize a sin table.
Builds on: unit-circle tip / symmetries (unit circle companion); formulas optional prior.
Tool gained: y = A sin(kx) (or cos) — amplitude |A|, period P = 360°/k from the turning tip.
- Principle. When a job repeats forever — a turning radius, a bobbing light — the height you care about must come back after one full turn. On a circle of radius 1, that height is sin x and the sideways run is cos x.
- Model. Write y = A sin(kx) (or cos). A stretches how tall the wave gets; k changes how fast the angle grows. Period P = 360°/k; amplitude |A| = (max − min)/2.
- Prediction / Answer. Larger |k| → shorter period. Read A and P from a sensor graph, or write the model for a bobbing beacon — same move, with a job.
1. Why a table is not enough
A common book move: dump a table of sin x values, draw a wiggle, and call it “the sine curve.” Missing: why the wiggle must look like that. Our rewrite: Fix a unit circle (radius 1). A radius turns through angle x. The tip’s height above the diameter is sin x; the sideways run is cos x. Plot height against the turn and you get a wave — the same picture that makes a held note a pressure ripple in sound, and the same rim you met in circle / Ring of Measure. Bitwave keeps only the sign of that height; here we keep the full height.
Links: sound · circle · Ring of Measure
Unit-circle height → sin / cos. On radius 1: height = sin x, sideways = cos x, for any turn — past 90° too. Negative height means below the diameter. That is the weapon that turns “periodic job” into a graph you can read.
2. Hands-on — spin, stretch, speed up
Drag amplitude A and speed factor k. The plot is y = A sin(kx) for one full cycle of the argument kx from 0° to 360°. Watch max height and how soon the wave repeats.
Period of the graph: P = 360° / k. Amplitude: |A| = (max − min) / 2.
A = 1 · k = 1 · P = 360° · |A| = 1
3. Period of sin x and cos x
After a full turn of 360°, the tip is back where it started — same height, same sideways. So y = sin x and y = cos x both repeat every 360°. Cos is the same engine: sideways offset instead of height; same period.
| x | sin x | cos x |
|---|---|---|
| 0° | 0 | 1 |
| 90° | 1 | 0 |
| 180° | 0 | −1 |
| 270° | −1 | 0 |
| 360° | 0 | 1 |
Table is a readout, not the reason. The circle is the reason.
4. Amplitude and the period tool
Amplitude |A|. For y = A sin x (or A cos x), the wave reaches +|A| and −|A|. Shortcut: |A| = (maximum − minimum) / 2.
Period P = 360° / k. For y = A sin(kx) or A cos(kx), the argument kx advances 360° when x advances 360°/k. Larger |k| → faster wiggle → shorter period.
5. Worked examples (reframed)
Example A — read A and P from a tide-like graph
Example A — read A and P from a tide-like graph. A harbour sensor logs height that peaks at +2.5 m and bottoms at −2.5 m. |A| = (2.5 − (−2.5)) / 2 = 2.5. Same arithmetic as the naked “1301”; we gave the sensor a job.
On a pure degree axis with P = 180°, k = 2. Core: |A| = 2.5, P = 360°/k.
Example B — write the model
Example B — write the model. A spinning shop sign’s tip height follows y = 1.5 cos(2x). Amplitude 1.5; k = 2 ⇒ P = 180° of shaft — two tip cycles per full spin.
Core facts unchanged: |A| = 1.5, k = 2, P = 360°/2 = 180°.
6. Fake cut tasks → real jobs
Exercises in the cut style (roughly “1303–1316”) often ask you to read A and k from a naked wiggle, or to grind sums like sin 1° + … + sin 359° with no story. Below: mark Fake, give a Real job, solve it.
Fake
“The graph of y = A sin(kx) is shown. Find A and k.” No units, no sensor, no question that needs the answer.
Real — day-length wave
At a latitude, daylight length over a year wobbles between 6 h and 18 h. Midline 12 h; amplitude (18−6)/2 = 6 h. One year ↔ one full cycle ⇒ if x is day-of-year mapped to 0°…360°, then k = 1 and y ≈ 12 + 6 sin(x − phase). You need A to plan batteries for a remote station.
How the fake becomes real
Same (max−min)/2 and same P = 360°/k. The real job supplies units and a reason to care about A.
Fake
“Compute sin 1° + sin 2° + … + sin 359°.” A calculator grind with no picture.
Real — symmetry on the circle
Pair height at θ with height at 360°−θ: they cancel (opposite sides of the diameter), and sin 180° = 0. The whole sum is 0 — because opposite rim heights cancel on a full turn. You learn the circle’s symmetry, not a keystroke marathon.
How the fake becomes real
The fake asks for a number; the real asks why the number must be zero. Tool: unit-circle height.
Fake
“State the period of y = 4 cos(3x).” Symbol in, symbol out.
Real — radio bob / pressure ripple
A beacon’s signal strength bobbles three times per antenna turn: model ∝ cos(3x). P = 360°/3 = 120° of shaft per bob. Designers need P to match a filter window — same formula, with a dial that matters.
How the fake becomes real
Weapon P = 360°/k applied to a spinning feed; k = 3 is “three bobs per turn,” not a random coefficient.
Fake
“Sketch y = −2 sin(x) without saying what −2 means in the world.”
Real — weather balloon on a line
A weather balloon bobs on its tether in wind. Take vertical displacement from the mean height as y = A sin(kx) with A from the photo scale and k from bob rate. Negative A (or a phase shift) just means you started the clock at a downswing — the |A| is still the weapon for “how hard does it yank the line?”
How the fake becomes real
The book’s decorative balloon becomes the sensor. Amplitude = yank strength; period = bob rate. Same A and k, now tied to a line you could instrument.
Gate — four, all required
Hands-on (move A or k) is 20. Each right gate item is 20. Gates ask you to rebuild and predict — not recite a table.
1. On the unit circle, sin x is best read as…
2. Why do y = sin x and y = cos x repeat every 360°?
3. A new beacon uses y = 2 cos(4x). Predict how many full signal bobs per 360° of shaft.
4. Amplitude |A| from a graph is…