Sine Calculator - Calculate Sin of Any Angle

The sine function, written sin(x), is one of the three primary trigonometric functions alongside cosine and tangent. In a right-angled triangle, the sine of an acute angle is defined as the ratio of the length of the side opposite that angle to the length of the hypotenuse. This ratio always lies between −1 and 1, regardless of the size of the triangle, which makes sine a dimensionless quantity perfectly suited for expressing proportions and periodic phenomena. The most intuitive way to extend the definition beyond acute angles is through the unit circle — a circle of radius 1 centred at the origin of the coordinate plane. Any angle measured from the positive x-axis corresponds to a point on the unit circle, and the sine of that angle is simply the y-coordinate of that point. As the angle increases from 0° to 90° the y-coordinate rises from 0 to 1; from 90° to 180° it falls back to 0; from 180° to 270° it drops to −1; and from 270° to 360° it returns to 0. This produces the characteristic smooth, repeating wave known as the sine wave, which has a period of 360° (or 2π radians). Angles can be measured in degrees or radians. A full circle is 360° or 2π radians, so to convert from degrees to radians you multiply by π/180, and to convert from radians to degrees you multiply by 180/π. Many scientific formulas — particularly in calculus, physics, and signal processing — use radians because the derivative of sin(x) in radians is simply cos(x), a clean result that does not hold when degrees are used. This calculator accepts both units and converts internally before computing. The sine function is periodic with period 2π radians (360°), which means sin(x + 2π) = sin(x) for all x. This periodicity is why sin(30°) = sin(390°) = sin(750°) = 0.5. The function is also odd, meaning sin(−x) = −sin(x), so negative angles simply flip the sign of the result: sin(−45°) = −sin(45°) ≈ −0.7071. Special angle values worth memorising: sin(0°) = 0, sin(30°) = 0.5, sin(45°) = √2/2 ≈ 0.7071, sin(60°) = √3/2 ≈ 0.8660, sin(90°) = 1, sin(180°) = 0, sin(270°) = −1. These arise from the geometry of 30-60-90 and 45-45-90 triangles. In practice, sine appears in an enormous range of applications. In physics, a pendulum's displacement, a vibrating string's shape, and the voltage of an AC circuit all follow sine curves. In signal processing and audio engineering, any complex periodic waveform can be decomposed into a sum of sine waves of different frequencies and amplitudes — this is the basis of Fourier analysis. In navigation and surveying, the law of sines (a/sin A = b/sin B = c/sin C) relates the sides and angles of any triangle. In computer graphics, sine and cosine are used together to compute rotations, generate circular motion, and create smooth animations. Modern calculators compute sine using efficient polynomial approximations derived from the Taylor series expansion: sin(x) = x − x³/3! + x⁵/5! − x⁷/7! + … where x is in radians. This series converges for all real numbers and is accurate to machine precision with relatively few terms near x = 0. For angles far from zero, implementations first reduce the angle to the range [−π/2, π/2] using the periodicity and symmetry properties of the function before applying the series. This calculator returns results accurate to ten significant decimal digits.