Calculate sin of any angle in degrees or radians with exact common values.
Common sine values
| Angle | Radians | sin(θ) | Decimal |
|---|---|---|---|
| 0° | 0 | 0 | 0.00000000 |
| 30° | π/6 | 1/2 | 0.50000000 |
| 45° | π/4 | √2/2 | 0.70710678 |
| 60° | π/3 | √3/2 | 0.86602540 |
| 90° | π/2 | 1 | 1.00000000 |
| 120° | 2π/3 | √3/2 | 0.86602540 |
| 135° | 3π/4 | √2/2 | 0.70710678 |
| 150° | 5π/6 | 1/2 | 0.50000000 |
| 180° | π | 0 | 0.00000000 |
| 270° | 3π/2 | −1 | −1.00000000 |
| 360° | 2π | 0 | 0.00000000 |
Definition and the unit circle
Right-triangle definition. In a right triangle, the sine of an acute angle θ is the ratio of the side opposite θ to the hypotenuse: sin(θ) = opposite ÷ hypotenuse.
Unit-circle definition. Draw a circle of radius 1 centred at the origin. Measure angle θ counter-clockwise from the positive x-axis. The point where the terminal side meets the circle has coordinates (cos θ, sin θ). So sin(θ) is simply the y-coordinate — defined for every real angle, positive or negative.
Range and period. The sine wave oscillates between −1 and 1, so −1 ≤ sin(θ) ≤ 1 for every θ. It repeats every 360° (2π radians): sin(θ + 360°) = sin(θ). It is an odd function: sin(−θ) = −sin(θ).
Frequently asked questions
How do I convert degrees to radians?
Multiply degrees by π ÷ 180. So 30° = 30 × π/180 = π/6 ≈ 0.5236 rad. To go back: radians × 180 ÷ π. Radians are the standard in calculus and physics because derivatives of sin and cos stay clean only when angles are in radians. Degrees stay more intuitive for geometry and everyday work.
Why is sin(30°) exactly 0.5?
Take an equilateral triangle with all sides equal to 1 and cut it along its height. You get a right triangle with hypotenuse 1, shortest side 1/2, and angles 30°–60°–90°. The side opposite the 30° angle is 1/2, so sin(30°) = (1/2) ÷ 1 = 1/2. The other classic exact values come from similar constructions: sin(45°) = √2/2 from an isosceles right triangle, sin(60°) = √3/2 from the same 30–60–90 triangle.
Can sine ever be greater than 1 or less than −1?
No. Geometrically, sin(θ) is the y-coordinate of a point on the unit circle, and that coordinate cannot leave the interval [−1, 1]. Any value outside this range is impossible for a real angle — which is also why arcsin(x) is only defined for x between −1 and 1.
Why does sin(180°) equal 0 and sin(270°) equal −1?
At 180° the terminal side of the angle points along the negative x-axis, so the y-coordinate is 0. At 270° it points straight down along the negative y-axis, so the y-coordinate is −1. The sine wave crosses zero every 180° and reaches its extremes (1 and −1) at 90° and 270°.
Where is sine used outside the classroom?
Anything that oscillates: alternating current (voltage follows V = V₀·sin(ωt)), sound waves, light waves, springs, pendulums, tides, and the Fourier analysis behind audio and image compression. In engineering and surveying, sine resolves a force or distance into components and handles triangulation. In navigation and astronomy it converts between angular and linear coordinates.
This sine calculator returns sin(θ) for any angle in degrees or radians, rounded to 8 decimal places, and shows the exact value whenever the angle is a standard one (0°, 30°, 45°, 60°, 90°, 180°, 270°, …). Switch between units with the tab, enter the angle — decimals and negative numbers are fine — and copy the result with one click. Examples: sin(30°) = 0.50000000 (exact 1/2); sin(45°) = 0.70710678 (exact √2/2); sin(60°) = 0.86602540 (exact √3/2); sin(90°) = 1; sin(180°) = 0; sin(π/4 rad) = 0.70710678. A reference table lists sine values for the classic angles from 0° to 360°, plus an expandable section covers the right-triangle and unit-circle definitions and the range [−1, 1] of the sine wave.