How this instrument works
The sine of an angle, sin(x), is the y-coordinate of the point that angle traces out on the unit circle — the circle of radius 1 centered at the origin. As the angle sweeps from 0° all the way around, sine smoothly rises and falls between −1 and 1, tracing out the familiar wave shape that gives the sine function its name.
In a right triangle, sine equals the side opposite a given angle divided by the hypotenuse — the SOH step in SOH-CAH-TOA, and geometrically the same y-coordinate reading, just scaled by whatever the triangle's hypotenuse happens to be instead of a fixed radius of 1.
Sine is periodic, repeating its exact same pattern every full 360° turn, and it's an odd function, meaning sin(−x) = −sin(x) for every angle — flipping an angle's sign flips its sine's sign too, a direct consequence of the unit circle's own symmetry across the x-axis.
- Enter an angle into the Angle field.
- Read sin(x): the sheet returns the corresponding value between −1 and 1.
- Try angles beyond 90° or negative angles to see the value cycle smoothly through its full range.
Worked example — sin(30°)
sin(30°) = 0.5 exactly — one of the small set of angles with a clean, memorizable sine value, worth recognizing rather than recomputing.
sin(0°) = 0 exactly, the trivial reference point where the unit circle sits right on the x-axis with no height at all. sin(90°) = 1 exactly instead — sine's own maximum, reached at a full quarter turn, where the corresponding point sits at the very top of the unit circle.
Questions
What does the sine function measure?
sin(x) is the y-coordinate of the point that angle x traces on the unit circle. In a right triangle, it equals the side opposite the angle divided by the hypotenuse — the SOH step in SOH-CAH-TOA.
What is the range of the sine function?
Always between −1 and 1, inclusive — sine can never output a value outside that band, no matter what angle is entered, since it's reading a coordinate off a circle of radius exactly 1.
Is sine periodic?
Yes — sine repeats its exact pattern every 360°, since sweeping a full turn around the unit circle returns to the identical starting point. sin(x) and sin(x + 360°) always give the identical result.
What does it mean that sine is an odd function?
sin(−x) = −sin(x) for every angle x — flipping the angle's sign flips the sine's sign too, a direct consequence of the unit circle's mirror symmetry across the horizontal axis.
How is this different from inverse sine?
Sine takes an angle and returns a ratio; inverse sine (asin) does the opposite, taking a ratio and returning the angle that produced it. This page computes the forward direction; the companion Inverse Sine page handles the reverse.