Math calculator

Arcsin Calculator

Q: What is arcsin?

Arcsin (arcsine) is the inverse of sine. It returns the principal angle whose sine equals the input x.

Q: What is the domain of arcsin?

For real numbers, x must be in [−1, 1]. Outside that interval the real arcsin is undefined.

Q: What range does arcsin return?

The principal value is in [−π/2, π/2] radians, or −90° to 90°.

Q: Is sin⁻¹ the same as 1/sin?

No. sin⁻¹(x) means inverse sine (arcsin), not the reciprocal (cosecant). The reciprocal is 1/sin(x) or csc(x).

Q: Is asin the same as arcsin?

Yes. asin is the common name in programming; arcsin and sin⁻¹ are the usual math notations.

Q: Why is arcsin(0) not also 180°?

Sine is periodic, so many angles share the same sine. Calculators return the principal value in [−90°, 90°]; other solutions differ by full periods or reflections.

Q: How is arcsin related to arccos?

For x in [−1, 1], arcsin(x) + arccos(x) = π/2 (90°). Also arcsin(−x) = −arcsin(x).

Calculate the principal arcsine in degrees and radians. Explore its graph, exact common angles, input limits, worked examples, and NIST reference sources.

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 · 2/2 declared partitions (arcsin, invalid-domain) · Matrix
Known limitations: x ∈ [−1, 1]; principal range [−90°, 90°]; nonzero canonical tiny deg/rad retained. Tabulated checks are not whole-domain assurance.
Core CVP does not include live graph, viewport, or pointer interaction.
Model: The principal real arcsine θ = arcsin(x) in degrees and radians. Default display is degrees.
Scope: Real inverse sine with principal branch [−π/2, π/2] (NIST DLMF §4.23(ii)). Complex arcsin is not returned.
Verification: Engine tested · Source checked · v1.0.8 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance· View Manifest · CVP overview · Specification
Versions: Calculation 1.0.8 · CVP protocol 1.0.0-proposed · Evidence 2026-10-16.inverse-trig-discovery-pins-v8
Verification revision: 2026-10-16.inverse-trig-discovery-pins-v8 · 5/5 property · digest 0d0f88919a7d
Legacy regression: 11/11 tests · Production surface contract 6/6
Trust layers: Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference: O1 model · O2 expected_values · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Interfaces: PASS · UI (SSR) / REST / MCP / URL→result→graph — Success 25/25. Error-path engine·REST·MCP 7/7 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only. Integration: URL → SSR result → Live graph current point (4/4). Hydration/slider/history are URL-canonical contracts, not a live browser session. Raw HTTP REST/JSON-RPC transport rejection 5/5; raw-number source and conflicting pins are transport-only checks, not engine/SSR input-source parity.
Supplemental domain review: Not performed
Named expert review: Not performed
CVP suite: 12/12 golden · 5/5 CVP boundary · 5/5 invalid · 5/5 property · 3/3 metamorphic · 5/5 round-trip · 19/19 O3 · 37/37 cross-interface · 4/4 URL→result→graph · 6/6 CVP contract · Manifest
Sources: NIST Digital Library of Mathematical Functions, Chapter 4 — Elementary Functions
NIST DLMF §4.23 — Inverse trigonometric functions; principal values
CalculatorX mathematical conventions — Principal branch; exact-value recognition; triangle mode is UI-only
Sources
Evidence: 6 legacy golden · 5 legacy boundary · legacy regression suite · 12/12 oracle-backed golden · 5/5 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.

Definitionθ = arcsin(x) = sin⁻¹(x)

Degreesθ° = arcsin(x) × 180/π

Right triangleθ = arcsin(opposite / hypotenuse)

iReal arcsin requires −1 ≤ x ≤ 1. The principal value lies in [−π/2, π/2]. General solutions are θ + 2kπ or π − θ + 2kπ.

How to use

Enter the sine value x

The published engine computes arcsin(x) for x in [−1, 1]. Opposite/hypotenuse is a local triangle convenience; share URLs use x.

Read degrees and radians

The main result is in degrees. Decimal radians sit under it; common values also show an exact π form.

Share the result

Share URLs include x, for example /calc/math/arcsin?x=0.5.

Example calculations

Common configurations with formula and result.

arcsin(0.5)

Classic acute angle

sin(30°) = 0.5

30° = π/6 rad

arcsin(1)

Right angle (principal)

sin(90°) = 1

90° = π/2 rad

arcsin(0)

Zero angle

sin(0°) = 0

0°

Opposite / hypotenuse

a = 6, c = 10

arcsin(6/10)

≈ 36.870°

Arcsin common values

Common values at a glance.

x	arcsin(x) °	arcsin(x) rad
−1	−90°	−π/2
−√3/2 ≈ −0.8660254	−60°	−π/3
−√2/2 ≈ −0.7071068	−45°	−π/4
−0.5	−30°	−π/6
0	0°	0
0.5	30°	π/6
√2/2 ≈ 0.7071068	45°	π/4
√3/2 ≈ 0.8660254	60°	π/3
1	90°	π/2

i Example: sin(30°) = 0.5 ⇒ arcsin(0.5) = 30° = π/6 rad.

Arcsin calculator specification

Version 1.0.8 · Engine tested

Calculation status

Engine tested 11/11 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.0.8

Review policy · Evidence

Definition: Arcsine is the inverse of sine. For x in [−1, 1], arcsin(x) is the unique angle θ in [−π/2, π/2] (−90° to 90°) such that sin(θ) = x. Notation: arcsin(x), asin(x), or sin⁻¹(x).
What it calculates: The principal real arcsine θ = arcsin(x) in degrees and radians. Default display is degrees.
Inputs: Real sine value x ∈ [−1, 1]
Outputs: deg: principal angle in degrees, range [−90, 90]
rad: the same angle in radians, range [−π/2, π/2]
Formula: θ = arcsin(x); θ° = θ × 180/π
Assumptions: Real inverse sine with principal branch [−π/2, π/2] (NIST DLMF §4.23(ii)). Complex arcsin is not returned.
Opposite/hypotenuse in the UI is a local triangle convenience (hypotenuse > 0, opposite ≥ 0, opposite ≤ hypotenuse). The published engine is arcsin(x). Use Decimal for a signed ratio, and Sin for the forward map.
When x is exactly a listed common value (0, ±1/2, ±√2/2, ±√3/2, ±1), the engine returns the exact algebraic angle such as 30° = π/6. Nearby floats are not snapped.
Units: Degrees (°)
Radians (rad)
Boundary conditions: Decimal strings are accepted and normalized to binary64 x. Nonzero decimal text that rounds to zero returns NUMERIC_UNDERFLOW; explicit zero remains valid.
REST/MCP accept only x and xInput; unknown fields return UNKNOWN_INPUT and conflicting aliases return AMBIGUOUS_INPUT.
Missing x returns MISSING_REQUIRED_INPUT.
Non-numeric, object, array, boolean, NaN, or Infinity inputs return INVALID_NUMBER.
x < −1 or x > 1 returns NOT_REAL (not clamped).
arcsin(−1) = −90°; arcsin(0) = 0°; arcsin(0.5) = 30°; arcsin(1) = 90°.
Numerical precision: Version pins reject unsupported API versions; shared pages use the current engine with an explicit mismatch notice. Historical execution and cross-runtime bitwise equality are not promised.
Computation uses IEEE-754 binary64 Math.asin. Canonical deg/rad are that numeric result unless x is bitwise-equal to a listed common value.
Exact algebraic angles are returned only when x === a listed common value.
Direct inputs are not repaired onto [−1, 1]. Values slightly above 1 are NOT_REAL.
A value within 1e-9 of a common x may include a display hint but does not replace deg/rad.
Nonzero canonical deg/rad values are preserved, including tiny angles; display rounding never replaces them with zero. REST and SSR return { deg, rad, verification }. The result card shows degrees as the primary value and decimal radians (plus an exact π label when x is a listed common value) as a secondary line.
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 θ = arcsin(x) using the same Full range / Focus template as Log (domain −1 ≤ x ≤ 1; vertical axis labelled θ (°)). Decimal mode has an engine-linked domain slider on [−1, 1]: dragging it writes x and updates the result, reverse check, and graph through the same local engine. Opposite/hypotenuse is a local triangle convenience: hypotenuse > 0, opposite ≥ 0, opposite ≤ hypotenuse.
Example: arcsin(0.5) = 30° = π/6
Validation cases: 8 published on this page · 11/11 tests · Production surface contract 6/6 · View evidence
x=0.5 → 30°
x=1 → 90°
x=0 → 0°
x=-1 → −90°
x=√2/2 → 45°
empty x → error MISSING_REQUIRED_INPUT
x=2 → error NOT_REAL
x=-0.5 → −30°
Sources: NIST Digital Library of Mathematical Functions, Chapter 4 — Elementary Functions
Supports: Inverse trigonometric functions as elementary inverses of sine, cosine, and tangent
NIST DLMF §4.23 — Inverse trigonometric functions; principal values
Supports: Principal arcsin on [−1, 1] with range [−π/2, π/2]; degree output is θ × 180/π
CalculatorX mathematical conventions — Principal branch; exact-value recognition; triangle mode is UI-only
Supports: Published engine is arcsin(x). Opposite/hypotenuse is a local triangle convenience (positive sides, opposite ≤ hypotenuse). Inputs outside [−1, 1] are NOT_REAL.
Calculation version: 1.0.8

Background

Interpretation and common distinctions.

What does this arcsin calculator compute?

Arcsine is the inverse of sine. For x in [−1, 1], arcsin(x) is the unique angle θ in [−π/2, π/2] (−90° to 90°) such that sin(θ) = x. Notation: arcsin(x), asin(x), or sin⁻¹(x).

The published formula or method is θ = arcsin(x); θ° = θ × 180/π. The following references support the stated mathematics or method:

NIST DLMF §4.23 — Inverse Trigonometric Functions: Inverse sine, cosine, and tangent, including principal branches.

Which inputs and limits should you check?

Use the real input domain [-1,1] and the principal inverse-sine branch. Read the displayed degree and radian values with their units.

The worked example uses Classic acute angle with sin(30°) = 0.5. 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.arcsin v1.0.8 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 arcsin is

Arcsine is the inverse of sine:

arcsin(x) = sin⁻¹(x)

If sin(30°) = 0.5, then:

arcsin(0.5) = 30° = (π)/6 rad

It answers: “which principal angle has this sine?”

Because sine is periodic, many angles share the same sine. This calculator returns the principal value in [−π/2,π/2].

Domain and range

	Real arcsin
Domain	x ∈ [−1, 1]
Range	[−π/2, π/2] rad = [−90°, 90°]

Right-triangle use

For a right triangle with opposite side a and hypotenuse c:

α = arcsin(a / c)

Example: a = 6, c = 10 → arcsin(0.6) ≈ 36.87°.

Identities

For x ∈ [−1, 1]:

arcsin(x) + arccos(x) = (π)/2

arcsin(−x) = −arcsin(x)

sin(arcsin(x)) = x

cos(arcsin(x)) = √(1 − x²), x ∈ [−1, 1]

Optional calculus forms:

Derivative: d/dxarcsin(x) = 1/(√(1−x²)) for |x|<1
Integral: intarcsin(x) dx = xarcsin(x) + √(1−x²) + C

Notation tip

sin⁻¹(x) means inverse sine, not 1/sin(x). The reciprocal of sine is csc(x). Prefer arcsin when you want to avoid that ambiguity.

Use Sin to go angle → sine, and Arccos / Arctan for the other inverse functions.

CVP (Calculator Verification Protocol)

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

Evidence Manifest: /evidence/math.arcsin/1.0.8.cvp.json
Independent reference: O1 (NIST DLMF model) + O3 precomputed mpmath tables (historical backend not inferred) for expected values and numerical behavior + O2 live identities
Numerical policy: IEEE-754 binary64; ≤2 ULP vs O3; declared_before_evaluation=true
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.
Expert review is optional and is not required for CVP Verified.

What happens with empty or extreme triangle lengths?

Enter both side lengths in the same unit. Empty lengths are errors, and a positive ratio that underflows to zero or overflows binary64 is rejected. The side-length convenience runs locally; shared links record the derived scalar x and reopen in Decimal mode.

How are request numbers and versions interpreted?

REST and MCP execute_calculation reject a nonzero raw JSON number such as 1e-400 with NUMERIC_UNDERFLOW before it becomes a zero calculation. Literal zero remains valid where the model permits it. Decimal strings preserve input text where declared, but do not extend binary64 precision; a value already sent as 0 cannot be recovered. exact: true identifies a recognized symbolic common-angle case for the normalized binary64 input. It does not promise lossless conversion of every source decimal or an exactly representable floating-point radian value.

Every supplied calculation-version pin must agree. Conflicting header/body pins return VERSION_PIN_CONFLICT (HTTP 409); the body does not override the header. See the request contract for accepted pin locations, input-source limits, and API contract 1.0.1. Calculation-model version and request-contract version are separate.

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

Frequently asked questions

Key distinctions behind the calculation.

What is arcsin?