Math calculator

Sin Calculator

Q: 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.

Q: 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.

Q: What is the range of sin?

Always between −1 and 1 inclusive.

Q: Why does sine repeat every 360°?

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

Q: How do I find the angle from a sine value?

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

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

Instant result

Result

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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 5/5
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: NIST Digital Library of Mathematical Functions, Chapter 4 — Elementary Functions
NIST DLMF §4.14 — Definitions and period of sine and cosine
CalculatorX mathematical conventions — Degree default; listed-identity exact values; inverse is Arcsin
Sources
Evidence: 7 legacy golden · 4 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 4/4 invalid · Artifact integrity PASS
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

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.

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.

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.

sin(30°)

Classic half

sin(π/6)

1/2 = 0.5

sin(90°)

Unit circle at (0, 1)

sin(π/2)

sin(0)

Root of sine

sin(0)

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°	0	0
30°	π/6	1/2
45°	π/4	√2/2
60°	π/3	√3/2
90°	π/2	1

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

Engine tested 11/11 tests · Production surface contract 5/5
Named expert review Not performed
Calculation version 1.0.5

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 5/5 · 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 does this sin calculator compute?

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).

The published formula or method is y = sin(θ_rad); θ_rad = θ° × π/180 when unit is deg. The following references support the stated mathematics or method:

NIST DLMF §4.14 — Trigonometric Functions: Definitions of sine, cosine, and tangent.

Which inputs and limits should you check?

Follow the selected angle unit and normalization policy. Degrees and radians are different representations of the same angle.

The worked example uses Classic half with sin(π/6). Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.sin v1.0.5 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

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.

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°].

Sin Calculator

Formulas

How to use

Enter the angle θ

Or use opposite / hypotenuse

Read sin(θ)

Example calculations

sin(30°)

sin(90°)

sin(0)

Opposite / hypotenuse

Sine common values

Sin calculator specification

Background

What does this sin calculator compute?

Which inputs and limits should you check?

How can you verify and cite this result?

What sine is

Degrees or radians

Periodicity and symmetry

Graph sketch

CVP (Calculator Verification Protocol)

Related tools

Frequently asked questions