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Math calculator

Sin Calculator

Calculate sin(θ) in degrees or radians. Fast local calculation with shareable, machine-readable results.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance
Input interpretation
Enter values to calculate.
Result
—
Assurance
Core
Declared partition coverage
PASS · 3/3 declared partitions (sin, invalid-domain, xcal) · Matrix
Numerical scope
≤2 ULP vs O3 applies to the 18 published tabulated vectors (common angles 0°…180° plus 8 seeded random angles in (−50°, 50°)). It is not a guarantee over the entire real line, including huge angles or unlisted neighborhoods of zeros.
Known limitations
  • deg|rad; textbook snap
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
The real sine y = sin(θ) for an angle in degrees or radians. Default unit is degrees.
Scope
Real sine. The published engine is forward sin(θ). Opposite/hypotenuse is a local triangle convenience (hypotenuse > 0, opposite ≥ 0, opposite ≤ hypotenuse). Use Arcsin for the inverse API.
Verification
Engine tested · Source checked · v1.0.5 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.5 · CVP protocol 1.0.0-proposed · Evidence 2026-09-08.xcal
Verification revision
2026-09-08.xcal · 3/3 property · digest f6d938207118
Legacy regression
11/11 tests · Production surface contract 3/3
Reference
O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP / URL→result→graph — Success 1/1. Integration: URL → SSR result → Live graph current point (4/4). Hydration/slider/history are URL-canonical contracts, not a live browser session.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
4/4 golden · 4/4 CVP boundary · 4/4 invalid · 3/3 property · 3/3 metamorphic · 4/4 round-trip · 18/18 O3 · 1/1 cross-interface · 8/8 cross-calculator · 4/4 URL→result→graph · 1/1 CVP contract · Manifest
Sources
Sources
Evidence
7 legacy golden · 4 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 4/4 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.5 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Degreessin(θ°) = sin(θ × π/180)
Radianssin(θ)
Trianglesin(θ) = opposite / hypotenuse
Unit circlesin(θ) = y-coordinate
iMost programming libraries take radians (e.g. SIN uses radians). This tool converts degree input before evaluating. Listed identities (0°, 30°, 45°, … mod 360°) return the algebraic value; nearby floats keep IEEE-754 Math.sin. Opposite/hypotenuse is a local triangle convenience; use Arcsin for the published inverse.

How to use

1

Enter the angle θ

Angle mode takes θ in degrees or radians. Share URLs include theta and unit. An engine-linked Overview range slider on [−360°, 360°] (or [−2π, 2π]) writes θ through the same local engine. Typed θ outside that window still calculates; the slider clamps and the caption marks outside Overview.

2

Or use opposite / hypotenuse

A local right-triangle convenience: hypotenuse > 0, opposite ≥ 0, opposite ≤ hypotenuse. Share URLs still use θ. Use Arcsin for the published inverse.

3

Read sin(θ)

Exact labels appear for listed identities (30°, 90°, …). The explainer reverse-checks with arcsin of the principal value in [−90°, 90°]; angles outside that range cite odd symmetry, 360° period, or the equivalent principal angle.

Example calculations

Common configurations with formula and result.

ϟ

sin(30°)

Classic half

sin(π/6)
1/2 = 0.5
ϟ

sin(90°)

Unit circle at (0, 1)

sin(π/2)
1
ϟ

sin(0)

Root of sine

sin(0)
0
ϟ

Opposite / hypotenuse

sin = a/c

1/2
1/2 (θ = 30° principal)

Sine common values

Common values at a glance.

θ (°)θ (rad)sin θ
−90°−π/2−1
−60°−π/3−√3/2
−45°−π/4−√2/2
−30°−π/6−1/2
0°00
30°π/61/2
45°π/4√2/2
60°π/3√3/2
90°π/21
i Example: sin(30°) = 0.5. Values repeat every 360° (2π rad). sin(π/6) = 0.5.

Sin calculator specification

Version 1.0.5 · Engine tested

Calculation status

Review policy · Evidence

Definition
Sine of an angle θ is the ratio of the opposite side to the hypotenuse in a right triangle, and the y-coordinate on the unit circle. Range is [−1, 1]. The function is odd and 2π-periodic: sin(−θ) = −sin(θ) and sin(θ + 360°) = sin(θ).
What it calculates
The real sine y = sin(θ) for an angle in degrees or radians. Default unit is degrees.
Inputs
  • Angle θ (finite real)
  • Optional unit: deg (default) or rad
Outputs
  • sin(θ) in [−1, 1]
Formula
y = sin(θ_rad); θ_rad = θ° × π/180 when unit is deg
Assumptions
  • Real sine. The published engine is forward sin(θ). Opposite/hypotenuse is a local triangle convenience (hypotenuse > 0, opposite ≥ 0, opposite ≤ hypotenuse). Use Arcsin for the inverse API.
  • Algebraic snap applies only when the reduced angle (mod 360°) is a listed identity such as 0°, 30°, 45°, 60°, 90°. Nearby floats, including 30.00000005°, keep IEEE-754 Math.sin.
Units
  • Dimensionless sine value
  • Degrees or radians for the angle
Boundary conditions
  • Missing θ returns MISSING_REQUIRED_INPUT. Invalid unit returns INVALID_UNIT. Malformed numbers, NaN, and ±Infinity return INVALID_NUMBER.
  • sin(0°) = 0; sin(30°) = 1/2; sin(90°) = 1; sin(150°) = 1/2; sin(180°) = 0; sin(−θ) = −sin(θ).
Numerical precision
  • IEEE-754 binary64 Math.sin after converting degrees with θ × π/180. That raw value is the published REST/SSR number unless θ reduces exactly to a listed identity.
  • Listed identities (0°, 30°, 45°, 60°, 90°, … mod 360°) return the algebraic value (so exact 180° is 0, not ~1e-16). There is no 1e-7° proximity snap.
  • Display may show ≈ 0 when |y| < 1e-12 and the input is not a listed identity. That rounding is UI-only and is not applied to the engine number.
  • The Result Card shows the degree → radian conversion next to the value (θ° = θ×π/180 rad) before sin(θ_rad). Shared URLs and REST use the same conversion.
  • The reverse check uses arcsin principal value in [−90°, 90°]. For negative θ outside that range it cites odd symmetry; for |θ| ≥ 360° it cites 360° period; for θ in (90°, 270°) it shows the equivalent principal angle. It never writes arcsin(y) ≈ θ when θ is outside [−90°, 90°].
  • Mathematical domain is all real θ. Numerical input is a finite IEEE-754 binary64 value. Extremely large finite angles can lose low-order phase bits in binary64 before 2π reduction; the engine still returns Math.sin of that converted argument and does not claim a unique mathematical angle.
  • Interactive evaluation runs in the browser. Shared URLs and REST use the same engine server-side.
  • The Result Card includes an engine-linked Live graph of y = sin(θ) using the same Overview / Focus template as Log (Overview window ±360° or ±2π; same degree/radian conversion as the engine). Angle mode has an engine-linked Overview range slider on [−360°, 360°] (or [−2π, 2π] in radians): dragging it writes θ and updates the result, reverse check, and graph through the same local engine. Typed θ outside Overview still evaluates; the slider clamps and the graph offers Focus. Opposite/hypotenuse is a local triangle convenience: hypotenuse > 0, opposite ≥ 0, opposite ≤ hypotenuse.
Example
sin(30°) = 0.5; sin(π/6 rad) = 0.5; sin(47.1°) ≈ 0.7325428987873787
Validation cases

30 published on this page · 11/11 tests · Production surface contract 3/3 · View evidence

  • theta=-360, unit=deg → 0
  • theta=-270, unit=deg → 1
  • theta=-180, unit=deg → 0
  • theta=-150, unit=deg → −1/2
  • theta=-90, unit=deg → −1
  • theta=-60, unit=deg → −√3/2
  • theta=-45, unit=deg → −√2/2
  • theta=-30, unit=deg → −1/2
  • theta=0, unit=deg → 0
  • theta=30, unit=deg → 0.5
  • theta=45, unit=deg → √2/2
  • theta=60, unit=deg → √3/2
  • theta=90, unit=deg → 1
  • theta=120, unit=deg → √3/2
  • theta=150, unit=deg → 0.5 (principal arcsin = 30°)
  • theta=180, unit=deg → 0
  • theta=210, unit=deg → −0.5 (principal arcsin = −30°)
  • theta=270, unit=deg → −1
  • theta=360, unit=deg → 0
  • theta=390, unit=deg → 0.5
  • theta=720, unit=deg → 0
  • theta=π/6, unit=rad → 0.5
  • theta=30.00000005, unit=deg → IEEE-754 Math.sin (not snapped to 1/2)
  • theta=47.1, unit=deg → 0.7325428987873787 (IEEE-754)
  • empty theta → error MISSING_REQUIRED_INPUT
  • theta=30, unit=foo → error INVALID_UNIT
  • theta=abc → error INVALID_NUMBER
  • theta=NaN → error INVALID_NUMBER
  • theta=Infinity → error INVALID_NUMBER
  • theta=1e16, unit=deg → IEEE-754 Math.sin after θ×π/180; low-order phase may be lost
Sources
  • NIST Digital Library of Mathematical Functions, Chapter 4 — Elementary Functions
    Supports: Sine as a standard trigonometric function with period 2π
  • NIST DLMF §4.14 — Definitions and period of sine and cosine
    Supports: sin(θ) on the real line; odd function; degree input converted via θ° × π/180
  • CalculatorX mathematical conventions — Degree default; listed-identity exact values; inverse is Arcsin
    Supports: Published engine is forward sin(θ) only. Algebraic snap is identity-only; |y|<1e-12 display rounding is UI-only.
Calculation version
1.0.5

Background

Interpretation and common distinctions.

What sine is

Sine is one of the six basic trigonometric functions. For an acute angle in a right triangle:

sin(θ) = opposite/hypotenuse

On the unit circle, sin(θ) is the y-coordinate after rotating by θ from the positive x-axis.

The result always lies in [−1, 1]. The period is 2π (360°), and sine is an odd function: sin(−θ) = −sin(θ).

Degrees or radians

Unit Typical use Conversion
Degrees (°) Geometry, surveying θ(rad) = θ° × π/180
Radians Calculus, most code / stats libraries θ° = θ(rad) × 180/π

Example: sin(π/6) = sin(30°) = 1/2.

Periodicity and symmetry

sin(θ + 360°) = sin(θ), sin(θ + 2π) = sin(θ)

sin(−θ) = −sin(θ)

Roots (zeros): θ = kπ for integer k.

Graph sketch

Plotting this calculator’s angle (in degrees) against sin(θ):

y = sin(θ)

  • Period 360° (2π); odd: sin(−θ)=−sin(θ)
  • At 0°, y=0; at 90°, y=1; at 180°, y=0

Use Arcsin for the published inverse.

CVP (Calculator Verification Protocol)

Math graph pilot under CVP 1.0 Proposed. Profile: core (pure sine transform, not an engineering model).

  • Evidence Manifest: /evidence/math.sin/1.0.5.cvp.json
  • Reproduce: /evidence/math.sin/reproduce — tabulated O3 inputs, expected values, generator, and re-run commands
  • Independent reference: O1 (NIST DLMF model) + O3 mpmath/MPFR expected values and numerical behavior + O2 live identities
  • Numerical policy: IEEE-754 binary64; ≤2 ULP vs O3 applies to the 18 published tabulated vectors (common angles 0°…180° plus 8 seeded random angles in (−50°, 50°)). It is not a guarantee over the entire real line, including huge angles or unlisted neighborhoods of zeros.declared_before_evaluation=true
  • Cross-calculator (CVP-XCAL-01): sin²θ + cos²θ = 1 vs published Cos; sin(arcsin x) = x vs published Arcsin for x ∈ (−1,1). Listed Tan / Arccos do not share those identities (N/A). Huge angles are N/A after binary64 phase-bit loss.
  • Interfaces: UI (SSR), REST, and MCP must agree. A separate integration check covers URL → result → Live graph. ui-ssr is not a live browser session.
  • Versions stay distinct: calculation 1.0.5 · CVP Proposed 1.0 · evidence revision 2026-09-08.xcal. Adding verification does not by itself change the sine algorithm.
  • Expert review is optional and is not required for CVP Verified.

Other calculators in this family: Arccos Calculator, Arcsin Calculator, Arctan Calculator, Cos Calculator, Pythagorean Theorem Calculator, Tan Calculator, Triangle Solver . Explore all Trigonometry & Triangles.

→ Arccos Calculator Calculate arccos(x) = cos⁻¹(x) for x in [−1, 1]. Fast local calculation with shareable, machine-readable results. › → Arcsin Calculator Calculate arcsin(x) = sin⁻¹(x) for x in [−1, 1]. Fast local calculation with shareable, machine-readable results. › → Arctan Calculator Calculate arctan(x) = tan⁻¹(x) for any real x. Fast local calculation with shareable, machine-readable results. › → Cos Calculator Calculate cos(θ) in degrees or radians. Fast local calculation with shareable, machine-readable results. › → Pythagorean Theorem Calculator Solve a right triangle with a² + b² = c². Enter any two of legs a, b and hypotenuse c to find the third side, plus area and perimeter. Try it free. › → Tan Calculator Calculate tan(θ) = sin(θ)/cos(θ) in degrees or radians. Fast local calculation with shareable, machine-readable results. › → Triangle Solver Solve a Euclidean triangle from SSS, SAS, ASA, AAS, or SSA (ambiguous case). Not the Pythagorean-only calculator. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

What is the sine function?

In a right triangle, sin(θ) = opposite / hypotenuse (acute θ). On the unit circle, sin(θ) is the y-coordinate after a rotation by θ.

Should I use degrees or radians?

Use the unit stated in your problem. Libraries and statistics software often expect radians; convert with θ_rad = θ° × π/180.

What is the range of sin?

Always between −1 and 1 inclusive.

Why does sine repeat every 360°?

Sine is periodic with period 360° (2π radians): sin(θ + 360°) = sin(θ).

Is sine an odd function?

Yes. sin(−θ) = −sin(θ).

What is sin(30°)?

Exactly 1/2 (0.5).

How do I find the angle from a sine value?

Use the Arcsin calculator for the principal value in [−90°, 90°].