Sine Calculator
Calculates the sine of an angle entered in degrees, returning a value between −1 and 1. Use it when solving right triangles, decomposing vector components, modeling waves, or working through trigonometry homework.
Last updated: May 2026
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About this calculator
The sine function is one of the three primary trigonometric ratios. For a right triangle, sin(θ) equals the length of the side opposite the angle divided by the hypotenuse: sin(θ) = opposite / hypotenuse. The unit-circle definition extends this to all angles: sin(θ) is the y-coordinate of the point on the unit circle (radius 1) at angle θ measured counter-clockwise from the positive x-axis. Because computers and most libraries (including JavaScript's Math.sin) work in radians internally, an angle given in degrees must first be converted using radians = degrees × π / 180. Variables: angle (in degrees, any real number — positive, negative, or beyond ±360°). Output is bounded between −1 and +1. Key reference values: 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. The function is periodic with period 360° (or 2π radians), meaning sin(θ + 360°) = sin(θ); it is also odd, meaning sin(−θ) = −sin(θ). Edge cases: very large angles wrap modulo 360° before computation; floating-point errors at exact multiples of π may produce values like 1.2246e−16 instead of 0 — round small results to zero in practice. Sine is widely used in physics for resolving force components, in engineering for AC signal analysis (waveforms), in geometry for the Law of Sines, and in computer graphics for rotations and animations.
How to use
Example 1: Find sin(45°). Step 1: convert to radians — 45 × π/180 ≈ 0.7854 radians. Step 2: compute sin(0.7854) ≈ 0.7071. Verify: in a classic 45-45-90 right triangle with legs of length 1 and hypotenuse √2, the opposite/hypotenuse ratio is 1/√2 = √2/2 ≈ 0.7071 — matches. Example 2: Resolve a 100 N force pulling at 30° above horizontal into its vertical component. Step 1: vertical = force × sin(angle) = 100 × sin(30°). Step 2: sin(30°) = 0.5 exactly. Step 3: vertical = 100 × 0.5 = 50 N. Verify: in a 30-60-90 right triangle, the side opposite the 30° angle is exactly half the hypotenuse — confirming sin(30°) = 0.5.
Frequently asked questions
What is the sine of an angle and how is it defined?
The sine of an angle θ is defined as the ratio of the side opposite that angle to the hypotenuse in a right triangle: sin(θ) = opposite / hypotenuse. This ratio is the same for all similar right triangles regardless of size. The unit-circle definition extends sine to all real numbers: sin(θ) is the y-coordinate of the point on the unit circle (radius 1) at angle θ. Sine values always fall between −1 and +1, repeating periodically with period 360°. The function is also useful in modeling oscillations: simple harmonic motion, alternating current, sound waves, and light waves are all expressed using sine.
Why does the sine calculator convert degrees to radians before computing?
Computers and mathematical libraries, including JavaScript's Math.sin(), calculate trigonometric functions using radians as their native unit. Radians are the mathematically natural angle measure, defined as arc length divided by radius — a full circle is 2π radians. The conversion formula is radians = degrees × π / 180, so 90° becomes π/2 ≈ 1.5708 radians and 180° becomes π ≈ 3.14159. This calculator applies the conversion automatically so you can simply type the familiar degree value. Calculus formulas for derivatives and integrals of trigonometric functions also assume radian inputs; using degrees there would require additional conversion factors and break the elegance of the underlying mathematics.
When should I use sine instead of cosine or tangent?
Use sine when you know an angle and the hypotenuse and need the opposite side, or when you know the opposite side and hypotenuse and need the angle. In physics, sine gives the perpendicular (vertical) component of a vector. For a force F at angle θ above horizontal, the vertical lift is F × sin(θ). Cosine is preferred for the adjacent side or horizontal component, while tangent is used when you know both legs and need the angle, or when you know an angle and one leg and need the other. The mnemonic SOH-CAH-TOA helps: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
What are common mistakes when computing sine values?
Forgetting to set your calculator or formula to degree mode (and computing sin(30) thinking 30 degrees when the engine interprets it as 30 radians) gives wildly wrong answers — sin(30 radians) ≈ −0.988, not 0.5. Confusing the inverse sine (arcsin or sin⁻¹) with the reciprocal (1/sin = cosecant) is a common notation error. Misapplying sine to non-right-triangle problems without using the Law of Sines or Law of Cosines leads to wrong results. Floating-point noise near exact zero values (sin(180°) = 1.22e−16 instead of 0) trips up condition checks — round near-zero results explicitly. For very large angles, mod by 360° first to avoid precision loss in the trig engine.
When should I NOT use a basic sine calculator?
Oblique (non-right) triangles need the Law of Sines (a/sin(A) = b/sin(B) = c/sin(C)) or Law of Cosines rather than a basic sine lookup. Hyperbolic sine (sinh) is a different function entirely, applicable to catenary curves, special relativity, and hyperbolic geometry — don't confuse the two. Spherical trigonometry on a sphere's surface uses different formulas involving the radius and great-circle distances. Solving for an angle when given the sine value requires the inverse function (arcsin), not the regular sine. For phase-shifted or amplitude-modulated waveforms (e.g., A·sin(ωt + φ)), you also need the amplitude A, angular frequency ω, and phase φ — a plain sine calculator only returns the unit-amplitude, zero-phase value. For complex-number angles or matrix-valued arguments, use specialized math libraries instead.