Arcsin Calculator

Definition

The inverse sine, written arcsin(x) or asin(x), is the inverse of the restriction of sine to an interval on which sine is one-to-one. The map sin : ℝ → [−1, 1] is periodic and is not injective on the whole real line: sin(θ) = sin(π − θ) = sin(θ + 2πk) for integers k. Hence no global single-valued inverse exists. On the closed interval [−π/2, π/2], sin is continuous and strictly increasing from −1 to 1, so it is bijective onto [−1, 1]. The inverse of that restriction is arcsin : [−1, 1] → [−π/2, π/2].

The range [−π/2, π/2] is the principal value. It is the standard branch in the NIST Digital Library of Mathematical Functions, §4.23, and in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 4. In degrees the principal range is [−90°, 90°]. This calculator uses the same branch: it computes arcsin in radians and, when requested, converts α° = α · (180/π).

Historical and conceptual context

Inverse trigonometric functions arose wherever an angle must be recovered from a sine value — in astronomy, surveying, and calculus. In modern analysis arcsin is one of the six standard inverse trigonometric functions. MathWorld treats Inverse Sine as multi-valued in the complex plane, with a distinguished real branch on [−1, 1].

Conceptually, arcsin does not select an angle outside the principal branch. For x = 1/2 both 30° and 150° have sine equal to 1/2, yet arcsin(1/2) = π/6. That choice is an analytic convention: it preserves oddness (arcsin(−x) = −arcsin(x)) and continuity on the whole domain. When a two-argument angle from (cos, sin) with quadrant resolution is required, atan2 is used rather than arcsin alone.

Formal properties

Domain: x ∈ [−1, 1]. A number outside this interval is not a real sine value, so no real result exists. Codomain (principal range): [−π/2, π/2]. The function arcsin is continuous on [−1, 1], strictly increasing, with arcsin(−1) = −π/2, arcsin(0) = 0, arcsin(1) = π/2. It is an odd function.

Identities for x ∈ [−1, 1]: sin(arcsin(x)) = x; arcsin(sin θ) = θ for θ ∈ [−π/2, π/2]; arcsin(x) + arccos(x) = π/2; arcsin(−x) = −arcsin(x); cos(arcsin(x)) = √(1 − x²) ≥ 0 on the principal branch.

Derivative: for |x| < 1, (arcsin x)′ = 1/√(1 − x²). At the endpoints the derivative has vertical asymptotes. Antiderivative: ∫ arcsin(x) dx = x·arcsin(x) + √(1 − x²) + C. These formulas match NIST DLMF §4.23 and Abramowitz & Stegun.

Full derivation of the branch and the derivative

Step 1. Let f(θ) = sin θ on [−π/2, π/2]. Then f′(θ) = cos θ ≥ 0 and is strictly positive on (−π/2, π/2), so f is strictly increasing. Since f(−π/2) = −1 and f(π/2) = 1, the image is [−1, 1].

Step 2. Therefore a unique inverse g : [−1, 1] → [−π/2, π/2] exists. Write g = arcsin. By definition, sin(arcsin(x)) = x and arcsin(sin θ) = θ on the indicated sets.

Step 3 (derivative). Let y = arcsin(x), so x = sin y. Differentiating in y yields dx/dy = cos y, hence dy/dx = 1/cos y. On (−π/2, π/2), cos y > 0 and cos y = √(1 − x²), so (arcsin x)′ = 1/√(1 − x²).

Step 4 (link to arccos). Let β = arccos(x) ∈ [0, π]. Then sin(π/2 − β) = x and π/2 − β ∈ [−π/2, π/2]. Uniqueness of arcsin gives arcsin(x) = π/2 − arccos(x), so arcsin(x) + arccos(x) = π/2.

Special cases

Standard values: arcsin(−1) = −π/2; arcsin(−√3/2) = −π/3; arcsin(−√2/2) = −π/4; arcsin(−1/2) = −π/6; arcsin(0) = 0; arcsin(1/2) = π/6; arcsin(√2/2) = π/4; arcsin(√3/2) = π/3; arcsin(1) = π/2. For x = 0.6 (the 3/5 ratio from a 3-4-5 triangle) the result is arcsin(0.6) ≈ 0.6435 rad ≈ 36.87°.

As x → ±1 the angle approaches ±π/2 and sensitivity (the derivative) grows without bound. Near |x| = 1 a small change in x produces a large change in angle — a geometric feature of the unit circle, not a calculator defect. For |x| > 1 no real result exists.

Numerical and computational notes

The browser uses arcsin (floating-point arithmetic, about 15–16 significant digits). In numerical literature the angle from (√(1−x²), x) is sometimes computed as atan2(x, √(1−x²)), which recovers the same [−π/2, π/2] branch and can be better conditioned near the endpoints. This calculator reports arcsin — the same principal value.

Input outside [−1, 1] is rejected with an error message; the calculator does not return a complex arcsin. A comma or a period in the field is accepted equivalently. Near |x| = 1, loss of precision in √(1−x²) in floating-point arithmetic is a known numerical caveat.

Related concepts and authorities

Related tools on this site: inverse cosine (range [0, π]), inverse tangent, sine of an angle (the forward map), and the degrees–radians converter. Arcsin does not compute triangle side lengths — use the law of sines for that.

Authorities: NIST DLMF §4.23; Abramowitz & Stegun, Chapter 4; MathWorld — Inverse Sine; Wikipedia — Inverse trigonometric functions (principal ranges). The calculator is educational and does not replace a textbook.