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 arcsin(x) = sin⁻¹(x) for x in [−1, 1]. Fast local calculation with shareable, machine-readable results.

Instant result

Result

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Enter values to calculate.

Inputs—

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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°]
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 · v1.0.5 · 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
CVP identity: 5/5 property · digest 8ccedfaab98f
Legacy regression: 11/11 tests · Production surface contract 3/3
Trust layers: Verification VERIFIED · Production CURRENT · overall VERIFIED
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: 6/6 golden · 5/5 CVP boundary · 5/5 invalid · 5/5 property · 3/3 metamorphic · 5/5 round-trip · 19/19 O3 · 1/1 cross-interface · 4/4 URL→result→graph · 3/3 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 · 6/6 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 1.0.5 · Engine tested

Calculation status

Engine tested 11/11 tests · Production surface contract 3/3
Named expert review Not performed
Calculation version 1.0.5

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: 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: 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.
Otherwise, |θ°| < 1e-12 is cleaned to 0. 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 3/3 · 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: 1.0.5

Background

Interpretation and common distinctions.

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/1.0.5.cvp.json
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; 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.

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

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Frequently asked questions

Key distinctions behind the calculation.

What is arcsin?