Sine, cosine, and tangent are not calculator buttons: they are lengths, measured from a floor someone chose. The choice is usually invisible — the horizontal is so common it seems natural. This is the instrument where it becomes visible, and where you can undo it with your hand.
Do: tilt the floor and see the entire apparatus rotate together. Move the radius and watch the two columns in the table: the lengths change, the ratios do not. Turn on the arc to compare sine, arc, and tangent by eye, and the wave to see the circle write the sine wave.
| length in the drawing | ratio (divided by the radius) | |
|---|---|---|
| cosine — along the floor | ||
| sine — perpendicular to the ground | ||
| tangent — on the tangent line | ||
| arc — from zero to the point | ||
| radius | 1 (by definition) |
1. The three are lengths, not magical numbers. The cosine is how far the point advanced along the ground; the sine, how far it rose from the ground; the tangent is measured along the line that touches the circle — and that's where it gets its name. They are the four lengths from Station 5 of C1.
1b. The T is not a loose point. The thin line that passes through the origin and point P continues and meets the tangent line: this meeting defines the T, and that's why the length of the tangent grows without limit as the radius approaches perpendicular to the floor — the two lines become increasingly parallel. And when the T goes out of the frame it does not disappear: the tangent is cut at the edge, with an arrowhead and the value displayed. Disappearing silently would hide exactly the asymptote — which is what we came to see.
And it pays more: with this line drawn, the two triangles that appear —
the small one, with sides cosine and sine, and the large
one, with sides radius and tangent — are similar. The ratio between
the sides is the same in both, and that is, as a drawing and not as a formula,
tan θ = sin θ / cos θ.
2. Tilt the floor. Nothing essential changes: the three lengths remain the same, rotated together. The angle was never "in relation to the horizontal" — it was always in relation to the floor we declared. The horizontal is the usual choice, and it's so habitual that it seems like the nature of things.
3. Adjust the radius, and look at the two columns of the table. The lengths change. The ratios do not. That's why the circle of radius 1 exists: in it, and only in it, the length is the ratio — and then you can call the drawing "sine" without lying.
4. Turn on the arc. With a radius of 1, the arc length is the angle in radians — that's the definition, not a conversion. And the three become comparable by eye: sine < arc < tangent. This inequality, obtained without any calculation, is what proves the limit that closes the series in C4.
Connect the wave and make it rotate. The height of the point — the sine — is moved to a position, and the curve is drawn as the angle moves: it's not a sine wave that appears ready, it's the circle writing one. Drag the angle with your hand and the trace advances and recedes with you. The dashed line is horizontal on purpose — it asserts one thing only: it's the same height, in two places.
Move the radius with the wave turned on. The amplitude is the radius — the wave does not have its own height, it has the height of the circle that generated it.
Why the floor locks. When you turn on the wave, the floor cursor resets to zero and locks. It's not a gimmick or oversight: the unrolled strip is nearly half a thousand pixels long, and it's impossible to rotate it within this frame without it leaving the screen — we measured. Unrolling requires choosing a floor, and here the choice is the usual one. Turn off the wave and its slope returns exactly as it was.
The bottom axis marks π and 2π for you to read the position, but it is compressed: it is not on the same scale as the radius. This sheet wants the image of the phenomenon; who makes the same passage in scale, with the arc and tangent together, is the next sheet — the unrolling.
Local, self-contained instrument: no network, no external library. From the same family as
par-vira-ponto.html and taxa.html.