Sin Calculator

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 v1.0.6 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

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/1.0.6.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; 121fixedvectorsdeclared_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.

Accepted inputs and precision

math.sin supports angle mode and the existing right-triangle ratio mode through the same local, SSR, REST and MCP engine. Angle mode uses finite binary64 θ with deg (default) or rad; aliases theta/t/angle/thetaInput and unit/unitInput must agree. Ratio mode uses finite opposite ≥0, hypotenuse >0 and opposite ≤hypotenuse; aliases opposite/oppInput and hypotenuse/hypInput must agree. Lengths need the same unit. Angle fields in ratio mode and side fields in angle mode are rejected. Unknown keys and duplicate shared-query keys are rejected. Missing mode defaults to angle; null/blank unit defaults to degrees. Empty or null numeric fields remain errors rather than default zero.

Numeric source text is strict decimal/scientific notation, at most 512 characters, without grouped numbers, hexadecimal or boolean coercion. A nonzero textual value that becomes zero in binary64 rejects NUMERIC_UNDERFLOW before computation. Raw JSON number literals receive the corresponding lossy-parse guard. Nonzero degree sine or nonzero side ratio that rounds to zero rejects RESULT_UNDERFLOW; finite nonzero subnormal results are preserved. Invalid results disable exports.

Degrees are reduced with a signed remainder before multiplication by π/180, then evaluated around a nearby quadrant. This avoids the loss introduced by adding360 near zero or by converting huge degrees first. Only listed degree identities and 17 specific normalized binary64 radian tokens within ±2π use symbolic recognition. The radian tokens are a documented symbolic convention, not mathematical exactness of floating-point π. Neighboring representable angles are not snapped. Generic radians use Math.sin of the accepted binary64 input. Results may legitimately round to ±1; relative sensitivity is unbounded near zeros, and large input spacing can lose intended phase before calculation.

The scalar result has a 2 ULP fixed-vector comparison budget with exact zero, sign and number type. The independent reference uses CPython Decimal420/620 Taylor series and Machin π plus Fraction ratios, never the production Math.sin. Both precisions must round to the same expected binary64. 121 independent vectors, strict comparator negative controls and invalid-input partitions cover published modes and boundaries; they do not prove the whole real line or cross-runtime bitwise agreement. All18 legacy O3 table values remain separately checked; previous1.0.5 pins and tables stay immutable. This is implementation verification, not third-party certification.

The main result normally uses12 significant digits, falling back to full binary64 text if that would change a nonzero or ±1 boundary. Full scalar and original inputs remain in native JSON/CSV. A ratio and any rounded derived angle are distinct; a derived angle is not independent validation of the ratio. Angle plots are omitted for |θ|>1e6 and in ratio mode; exact input text and numerical results remain available. The plot is a visualization, not an independent reference.

Typing computes locally. Shared URLs and REST/MCP send input parameters and may enter browser/server logs; no zero-log or whole-site network privacy guarantee is made. Mathematical definitions and series: NIST DLMF trigonometric definitions and DLMF power series.