How this instrument works
Sine is the ratio of a right triangle's side opposite an angle to its hypotenuse, but the definition that keeps working past 90° comes from a circle of radius 1 centered on the origin: sweep an angle θ counterclockwise from the positive x-axis, and sin(θ) is the height — the y-coordinate — of the point you land on. The two pictures agree everywhere the triangle survives to be drawn; the circle version simply keeps producing an answer once the triangle collapses.
Sweep the angle the other way, into negative territory, and sine responds by flipping sign rather than holding still: sin(−θ) = −sin(θ), a property called oddness. Geometrically it happens because a clockwise sweep by the same amount lands the point exactly as far below the x-axis as the counterclockwise sweep put it above — a reflection through the origin, not a mirror across an axis. That single fact is why a sine graph threads through the origin on its way up, while its cosine counterpart peaks there instead.
The name itself is an accident of translation. Indian astronomers called this ratio jya-ardha, 'half-chord'; Arabic translators wrote the sound of the word as jiba, and later copyists, reading an unpointed script where vowels are left out, misread that as jaib, an ordinary word for 'bay' or 'fold in a garment.' Latin translators rendered jaib literally as sinus, a fold or curve, and sinus is where 'sine' comes from — mathematics carrying a centuries-old transcription slip as its permanent name. Near θ = 0 the approximation sin(θ) ≈ θ (θ in radians) holds so tightly that pendulum clocks and small optical deflections are routinely modeled with it, swapping a transcendental function for the angle itself.
- Type your angle into the Angle, θ field — the unit selector next to it defaults to degrees, so a plain 30 means 30°.
- Switch to radians or turns if that is how your angle arrived; the sheet converts internally before it evaluates anything.
- Read sin(θ) for the result — it always lands between −1 and 1, with the sign marking whether the swept point sits above or below the x-axis.
- Check θ = 30°, 90°, and 150° as landmarks: sin(θ) should read 0.5, 1, and 0.5 again, since 30° and 150° sit at the same height on the circle.
- Enter a negative angle to see the odd-function property firsthand: sin(−30°) reads −0.5, the mirror image of sin(30°) through zero.
Worked example — the 30° reference angle
Set Angle, θ to 30 with the unit selector on degrees. Internally the sheet converts that to 0.5235987755982988 radians — exactly π ⁄ 6 — before evaluating, and sin(θ) returns 0.49999999999999994, which the display rounds to a clean 0.5: floating-point arithmetic leaves that trailing 4 behind, but the mathematics is exact, since 30° is one of the handful of angles where sine has a tidy closed form.
The result checks itself against a 30-60-90 triangle with hypotenuse 2: the side opposite the 30° angle measures 1, so opposite ⁄ hypotenuse = 1 ⁄ 2 = 0.5, matching the sheet exactly. The Pythagorean identity agrees too — cos(30°) ≈ 0.8660254 — since 0.5² + 0.8660254² = 0.25 + 0.75 = 1.0 exactly, confirming the point sits precisely on the unit circle's rim.
Questions
What does sin(θ) actually measure?
It's the height — the y-coordinate — of the point where an angle θ, swept counterclockwise from the positive x-axis, meets a circle of radius 1. In a right triangle the same ratio is the side opposite the angle divided by the hypotenuse; both descriptions agree wherever a triangle can be drawn, but the circle definition keeps working past 90°, where no triangle survives.
Why is sin(θ) always between −1 and 1?
Because it's a coordinate on a circle of radius 1 — the swept point can never sit farther than one unit from the center, so its height can never exceed that radius. The two extremes, sin(θ) = 1 and sin(θ) = −1, occur at θ = 90° and θ = 270°, the top and bottom of the circle.
How is sine different from cosine?
Sine and cosine are the same underlying wave, staggered against each other so that sine reaches zero exactly where cosine reaches its peak, and vice versa; formally, sin(θ) = cos(90° − θ). On the unit circle, sine tracks how high the swept point sits while cosine tracks how far right it sits. The distinction that matters for algebra is parity: sine is odd, so sin(−θ) = −sin(θ), whereas cosine is even and cos(−θ) = cos(θ).
What's the most common mistake people make with sin⁻¹?
Reading sin⁻¹ as 'one divided by sin' rather than 'the angle whose sine is this.' sin⁻¹(x) is arcsine, the inverse function, while 1 ⁄ sin(θ) is cosecant, a different quantity entirely. The notation is genuinely ambiguous outside calculus texts, which is why many calculators label the inverse button asin instead. This sheet only computes sin(θ) forward, from angle to ratio, so the mix-up cannot occur here.
What is sin(θ) at the standard reference angles?
sin(0°) = 0, sin(30°) = 1⁄2 exactly, sin(45°) = √2⁄2 ≈ 0.7071, sin(60°) = √3⁄2 ≈ 0.8660, and sin(90°) = 1 — the same five values as cosine's reference angles, read in reverse order, a quick way to check a table and this sheet at once.
Why does sin(θ) ≈ θ work for small angles?
Because near θ = 0 the unit circle's arc and the straight vertical drop to the x-axis are nearly identical in length — the curve hasn't had room to bend away from the tangent line yet. In radians, sin(θ) and θ differ by less than 0.05% for angles under about 3°, and by only about 0.13% even at a full 5°, which is why pendulum periods and small-signal optics quietly substitute θ for sin(θ) and lose almost nothing.