Sine Calculator

What is Sine?

In a right triangle, the sine of an angle θ is the ratio of the opposite side to the hypotenuse: sin(θ) = opposite / hypotenuse. Its values always fall between −1 and 1.

A few notable values are worth memorizing: sin(0°) = 0, sin(30°) = 0.5, sin(45°) ≈ 0.7071, sin(60°) ≈ 0.8660 and sin(90°) = 1.

Unit conversions: 1° equals π/180 rad ≈ 0.01745 rad, while 1 rad equals 180°/π ≈ 57.296°. And 400 gradians complete a full 360°.

Sine function: definition and example

In a right triangle, sin(θ) = opposite / hypotenuse. Extended to the unit circle, sin(θ) is the y-coordinate of the point at angle θ from the positive x-axis. It is periodic with period 2π (360°), odd — sin(-θ) = -sin(θ) — with amplitude 1. Notable values: sin(0) = 0, sin(30°) = 1/2, sin(45°) = √2/2, sin(60°) = √3/2, sin(90°) = 1. Example: for θ = 30°, sin(30°) = 0.5.

The Taylor series gives sin(x) = x - x³/3! + x⁵/5! - x⁷/7! + ... (radians) — how calculators compute it. For any triangle, the law of sines holds: a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius.

Applications

Sine describes every wave phenomenon: light, sound, electromagnetic radiation, tides. Alternating current is modelled as i(t) = I₀·sin(ωt + φ). GPS uses sines in multilateration to triangulate position from satellite distances. In music, timbre is the Fourier sum of sines at integer multiples of the fundamental. In computer graphics, sine drives rotations and animation curves. Sundials and astronomical navigation depend on sines of solar altitude.