How this instrument works
Picture a point traveling around a circle of radius 1, centered at the origin, starting from the positive x-axis and rotating counter-clockwise by a chosen angle. That point's y-coordinate — its height above or below center — is, by definition, the sine of that angle. This is the same sine function used in right-triangle trigonometry, just viewed from the unit-circle perspective rather than the triangle-ratio one.
Framed this way, sine's whole behavior becomes visually obvious: it starts at 0 (the point begins right on the x-axis), climbs to a maximum of 1 a quarter of the way around (the very top of the circle), drops back through 0 at the halfway point, bottoms out at −1 three-quarters around, and returns to 0 after a full turn — a smooth, repeating wave traced out by nothing more than a point circling a wheel.
This page accepts angles in degrees, radians, or full turns interchangeably, since the unit-circle picture doesn't care which unit describes the rotation — only how far around the circle the point has actually traveled.
- Enter an angle into the Angle field, in whichever unit is most convenient.
- Read y-coordinate on the unit circle: that angle's sine value.
- Try 90° to see the point sitting at the very top of the circle, y=1.
- Try 270° to see the point at the very bottom, y=−1.
Worked example — 30°, 90°, and 270°
At 30° around the unit circle, the rotating point has a y-coordinate of exactly sin(30°)=0.5 — already slightly above center, a third of the way to the top.
At 90°, the point sits at the very top of the circle, y-coordinate exactly 1 — sine's maximum possible value. At 270°, the point sits at the very bottom, y-coordinate exactly −1 — sine's minimum possible value, three-quarters of the way around a full turn.
Questions
What does sine represent on a unit circle?
The y-coordinate (height above or below center) of a point rotating counter-clockwise around a circle of radius 1, starting from the positive x-axis — sine at any angle is simply that point's vertical position at that point in the rotation.
Why does sine repeat every full turn?
Because a point rotating around a circle returns to its exact starting position after one full revolution, and its y-coordinate returns with it — sine is a periodic function precisely because circular motion itself repeats.
What is sine's maximum and minimum value?
1 (at the very top of the unit circle, a quarter turn around) and −1 (at the very bottom, three-quarters around) — sine never exceeds those two bounds, since the circle's own radius never exceeds 1.
Does it matter which angle unit I use?
No — degrees, radians, and full turns all describe the identical rotation around the circle, just measured on different scales; the resulting y-coordinate (sine value) comes out the same regardless of which unit was entered.
How is this related to right-triangle trigonometry?
It's the same sine function, just viewed differently: a right triangle's opposite-over-hypotenuse ratio and the unit circle's y-coordinate describe the identical relationship, since a unit circle's radius (always 1) plays the role of the hypotenuse.