Sine Calculator
Calculate the sine of an angle in degrees, radians or gradians. Precise result with up to 10 significant digits.
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.
FAQ
Why is sin(90°) = 1? On the unit circle, 90° points straight up; its y-coordinate is 1 — the maximum the function ever reaches.
Degrees, radians or grads? Math identities and calculus use radians (π rad = 180°). Degrees are the everyday unit. Grads (gon, 400 in a full turn) are used in some surveying and European engineering.
How does it relate to cosine? They are the same wave shifted by 90°: sin(θ) = cos(θ - 90°). The identity sin²θ + cos²θ = 1 follows from Pythagoras on the unit circle.
Can sine exceed 1? No — for real angles, sin(θ) lies in [-1, 1]. It can exceed 1 only for complex arguments (hyperbolic regime).
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Calculate the sine of an angle
The sine is one of the fundamental trigonometric functions. It shows up in triangle geometry and stays present in the description of waves and periodic motion. This calculator finds the sine of any angle, with a precise result of up to ten significant digits.
There's one detail that makes a difference: it accepts the angle in degrees, radians or gradians, and choosing the right unit avoids the most common mistake in trigonometry. You enter the angle, choose the unit and get the sine value, with no need for a table or a scientific calculator.
The calculation is done in the browser. It works for maths and physics exercises, engineering projects or any calculation involving this function.