Math / Turning on a circle
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Going around
Turning on a circle
A point walks the rim of a circle of radius 1 (enhetscirkeln). From the centre, its sideways offset is cosine, its height is sine. Past 90° the same names still work — height can fall, sideways can go negative. The circle is the picture that makes angles bigger than a right corner make sense as a job, not a slogan.
Origins
Ptolemy
Three points first: centre + two rim points. The angle picks the pair; the table’s number was the straight — not the arc. Half-chord ÷ radius became . Unit circle: radius 1, so the coordinates are the ratios.
1700s
Writing sin and cos as the coordinates of the point at angle θ on the unit circle made past-90° and negative values ordinary, not special cases to memorise.
— rim length
A full turn has rim length 2π on a circle of radius 1. Half a turn is π. A right corner is π/2. Radians are “how much rim,” not a definition dump — useful when the turn itself is the length you care about (radianer).
Honesty: we draw one circle. We are not selling a poster of every identity.
Drag the point around the rim
Radius is 1. The point’s x is cos(θ), its y is sin(θ). Drag past 90° — height falls back toward zero while the turn keeps growing. That is why the names still work after a right corner (sinus / cosinus som koordinater).
θ = 35° · cos ≈ 0.82 · sin ≈ 0.57
Soft angle: rim from the positive x-axis to the point is θ in radians when radius is 1. At 180° the rim is π ≈ 3.14 — half the full loop.
Enhetscirkeln: radie 1. cos θ = x, sin θ = y. Radianer: båglängd på enhetscirkeln.
Three points: centre, rim, rim
Draw the centre and two points on the rim. Two radii make the central angle. The straight cut between the rim points is the — that was the table’s number, not the curved arc. Split the chord in half: half-chord ÷ radius is .
The chord is not a side of the little right triangle used for sin. That triangle uses the radius, the drop to the diameter, and the run along the diameter. The chord is a separate line joining two rim points.
Central angle 70° · chord (straight) · half-chord → sine
Thick: chord (rim to rim). Dashed drop: the sin triangle’s height — a different line. Half the chord equals that height when the chord is centred on the diameter.
Tre punkter: mittpunkt + två på randen. Kordan är rak; bågen är böjd. Halva kordan / radie → sinus.
Signed height — below the diameter
At 0° and 180° the tip sits on the diameter. The right-triangle construction flattens to a line — a limiting case. The circle still names a height of 0. Drag the rim point through those angles above; watch sin cross zero.
Same circle, signed height. The wave later is just that signed number as the turn keeps going.
Why past 90° still has a job
Same names. Bigger picture. The triangle ratios were a slice of the circle; the circle keeps going.
Soft angles — how much rim
Degrees cut a full turn into 360 equal wedges — handy for maps and old instruments. Radians ask: on a circle of radius 1, how long is the arc you walked? Full turn → 2π. Half → π. Right corner → π/2. When the turn itself is a length (spinning wheels, phase in a wave), rim-measure fits the job better than “out of 360.”
Next wall — let the angle keep turning
You dragged one freeze at a time. Let the turn keep going in time: the height of the tip draws a wave. That plot is why a held piano note is not a flat push of air — why a note is a wave (sinusvåg · ljud). Unlocks when this lesson hits 100%.
Gate — four, all required
Hands-on (drag the turn) is 20. Each right gate item is 20.
1. On the unit circle, what are sine and cosine?
2. You drag past 90°. Height starts falling. What does that mean?
3. A radian is best thought of as…
4. Which is a real reason to leave the right-triangle picture?