Arcsine Calculator (arcsin)

Definition

Arcsin gives the angle y in the principal interval such that sin(y) = x. Equivalently, it's the inverse of sin restricted where sin is monotone.

y = \arcsin(x), \quad x \in [-1, 1], \quad y \in [-\tfracπ2, \tfracπ2]

Domain, range, principal branch

Because sin takes values in [−1, 1], arcsin is only defined for x in that interval. Outside it, arcsin is undefined in the reals — it returns a complex angle.

\mathrmdom(\arcsin) = [-1, 1], \quad \mathrmrange(\arcsin) = [-π/2, π/2]

Symmetry and identities

Arcsin is odd and has a useful co-function identity with arccos:

\arcsin(-x) = -\arcsin(x), \qquad \arcsin(x) + \arccos(x) = \tfracπ2

Common values

A handful of values are worth memorizing:

\arcsin(0) = 0 \quad \arcsin(\tfrac12) = \tfracπ6 \quad \arcsin(\tfrac√(2)2) = \tfracπ4 \quad \arcsin(\tfrac√(3)2) = \tfracπ3 \quad \arcsin(1) = \tfracπ2

Numerical evaluation

For other inputs, arcsin(x) is computed via Taylor series or via elementary identities. Most calculators and languages use a numerically stable series or a continued-fraction expansion.

Domain, Range and the Principal Value

The sine function only ever produces outputs between −1 and 1, so arcsine only accepts inputs in that interval. An input outside [-1, 1] has no real answer at all — a genuinely undefined result rather than a very large one, which is why calculators return an error rather than a number.

Sine is also not one-to-one: infinitely many angles share the same sine, since \sin 30° = \sin 150° = \sin 390° = 0.5. To make an inverse function possible at all, the range is restricted to the principal branch [-90°, 90°], or [-π/2, π/2] in radians. So arcsin(0.5) returns 30° and nothing else. This restriction is a convention, not a property of the mathematics, and it is the single most common source of confusion when using the function.

\arcsin: [-1, 1] \to [-(π)/(2), (π)/(2)]

Recovering the Other Solutions

Because the principal value discards valid answers, solving a real equation usually requires putting them back. For \sinθ = x the complete solution set is θ = \arcsin x + 360°n together with θ = 180° - \arcsin x + 360°n, for any integer n.

This matters whenever the physical angle can be obtuse. A projectile launched to hit a given range has two possible launch angles — a flat, fast trajectory and a lofted one — and they are exactly this pair, summing to 90° in that particular case. In the law of sines, the "ambiguous case" of triangle solving is the same phenomenon: knowing two sides and a non-included angle can leave two valid triangles, and only checking both candidates reveals which fits the remaining constraints. Always ask whether the context permits an angle beyond the principal range before accepting a single answer.

θ = \arcsin x + 2π n \quad \textor \quad θ = π - \arcsin x + 2π n

Useful Identities and Applications

Several identities make arcsine easier to work with. It is an odd function, \arcsin(-x) = -\arcsin(x). It pairs with arccosine through \arcsin x + \arccos x = π/2. And drawing the defining right triangle gives \cos(\arcsin x) = √(1-x^2) and \tan(\arcsin x) = x/√(1-x^2), which is how expressions mixing inverse and forward trigonometric functions get simplified.

The derivative 1/√(1-x^2) is why arcsine appears throughout integral calculus as the antiderivative of that expression, and it also shows the function steepening without bound as x approaches ±1 — small measurement errors near the endpoints produce large angle errors. In practice arcsine turns ratios back into angles: finding the angle of a ramp from its rise and slope length, applying Snell's law to refraction, computing the critical angle for total internal reflection, or recovering a phase from a normalised amplitude.

(d)/(dx)\arcsin x = (1)/(√(1-x^2)), \qquad \arcsin x + \arccos x = (π)/(2)

Worked Example

Inputs

• value: 0.5

Result: angleRad: 0.5236 angleDeg: 30 complementaryDeg: 60

Explanation

arcsin(0.5) = π/6 ≈ 0.5236 rad = 30°. The complementary angle (with arccos) is π/3 = 60°.

Second Scenario

Inputs

• value: 1

Result: angleRad: 0.5236 angleDeg: 30 complementaryDeg: 60

Explanation

This scenario uses different inputs (value = 1) to show how changing one variable affects the arcsine calculator (arcsin) result. Run the calculator above with these values to get the exact updated output with step-by-step work.