The Circle, Unrolled
Drag the point all the way around the circle. A dashed line connects it to the sine curve at the exact same height — the curve is the circle's own height, at every angle, read off sideways.
θ = 40°, sin θ = 0.64
Drag the point all the way around the circle.
- Angle θ
- 40°
- sin θ
- 0.64
- Point's height (y)
- 0.64
Jump to an angle
Ready to try some? Exact trig values worksheets
Where it usually goes wrong
sin(40°) is produced by a calculator button and treated as a fact retrieved from nowhere in particular — a table entry, or a black box — rather than a length that already exists on a picture a student may have drawn themselves an hour earlier. The unit circle and the sine graph are taught as two different representations without anyone showing they are the SAME number at the SAME moment.
Try this
- Drag the point slowly from 0° up to 90°. Watch the dashed connector line stay perfectly horizontal the entire time — it has to, because both ends are defined to be at the same height.
- Keep dragging past 90° toward 180°. The circle point drops back down, and the curve's dot drops with it, in the same instant — nothing is computed separately for the second half of the swing.
- Press 270°. The circle point is now at the very BOTTOM of the circle, and the curve's dot is at its lowest point too — sin is negative here for exactly the reason the point is below the center line, not because of a sign rule to remember.
Where this goes next
This is why sine repeats every 360°: dragging the point all the way around brings it back to where it started, so the curve is DESTINED to repeat too — periodicity is a property of walking around a circle, not a separate fact bolted onto the graph.